Loogle!

Result

Found 169 declarations mentioning LinearIndepOn.

LinearIndepOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} (R : Type u_2) {M : Type u_4} (v : ι → M) [Semiring R] [AddCommMonoid M] [Module R M] (s : Set ι) : Prop
linearIndepOn_empty 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} (R : Type u_2) {M : Type u_4} (v : ι → M) [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndepOn R v ∅
LinearIndepOn.of_subsingleton 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Subsingleton R] : LinearIndepOn R v s
LinearIndepOn.injOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] (hv : LinearIndepOn R v s) : Set.InjOn v s
LinearIndependent.linearIndepOn_univ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndependent R v → LinearIndepOn R v Set.univ
linearIndepOn_univ_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndepOn R v Set.univ ↔ LinearIndependent R v
LinearIndependent.linearIndepOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (h : LinearIndependent R v) (s : Set ι) : LinearIndepOn R v s
LinearIndependent.linearIndepOn_id 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (i : LinearIndependent R v) : LinearIndepOn R id (Set.range v)
LinearIndependent.of_linearIndepOn_id_range 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} {f : ι → M} (hf : Function.Injective f) : LinearIndepOn R id (Set.range f) → LinearIndependent R f
linearIndepOn_id_range_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} {f : ι → M} (hf : Function.Injective f) : LinearIndepOn R id (Set.range f) ↔ LinearIndependent R f
LinearIndepOn.linearIndependent_restrict 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (hs : LinearIndepOn R v s) : LinearIndependent (ι := ↑s) R (s.domRestrict v)
linearIndependent_restrict_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndependent (ι := ↑s) R (s.domRestrict v) ↔ LinearIndepOn R v s
LinearIndependent.linearIndepOn_id' 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (hv : LinearIndependent R v) {t : Set M} (ht : Set.range v = t) : LinearIndepOn R id t
LinearIndepOn.mono 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {t s : Set ι} (hs : LinearIndepOn R v s) (h : t ⊆ s) : LinearIndepOn R v t
linearIndepOn_iff_image 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} {s : Set ι} {f : ι → M} (hf : Set.InjOn f s) : LinearIndepOn R f s ↔ LinearIndepOn R id (f '' s)
LinearIndependent.of_linearIndepOn_range 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} {f : ι → ι'} (hf : Function.Injective f) (g : ι' → M) : LinearIndepOn R g (Set.range f) → LinearIndependent R (g ∘ f)
linearIndepOn_range_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type u_6} {f : ι → ι'} (hf : Function.Injective f) (g : ι' → M) : LinearIndepOn R g (Set.range f) ↔ LinearIndependent R (g ∘ f)
LinearIndepOn.linearIndependent 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι} (h : LinearIndepOn R v s) : LinearIndependent R fun (x : ↑s) => v ↑x
linearIndependent_set_coe_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : (LinearIndependent R fun (x : ↑s) => v ↑x) ↔ LinearIndepOn R v s
linearIndependent_subtype_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set M} : LinearIndependent R Subtype.val ↔ LinearIndepOn R id s
linearIndependent_comp_subtype_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndependent (ι := ↑s) R (v ∘ Subtype.val) ↔ LinearIndepOn R v s
LinearIndepOn.zero_notMem_image 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] (hs : LinearIndepOn R v s) : 0 ∉ v '' s
LinearIndepOn.ne_zero 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] {i : ι} (hv : LinearIndepOn R v s) (hi : i ∈ s) : v i ≠ 0
linearIndepOn_zero_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] : LinearIndepOn R 0 s ↔ s = ∅
linearIndepOn_iff_linearIndepOn_finset 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndepOn R v s ↔ ∀ (t : Finset ι), ↑t ⊆ s → LinearIndepOn R v ↑t
LinearIndepOn.singleton 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} {i : ι} [IsDomain R] [Module.IsTorsionFree R M] (hi : v i ≠ 0) : LinearIndepOn R v {i}
linearIndepOn_equiv 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (e : ι ≃ ι') {f : ι' → M} {s : Set ι} : LinearIndepOn R (f ∘ ⇑e) s ↔ LinearIndepOn R f (⇑e '' s)
LinearIndepOn.notMem_span_of_insert 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {i : ι} [Nontrivial R] (hv : LinearIndepOn R v (insert i s)) (hi : i ∉ s) : v i ∉ Submodule.span R (v '' s)
LinearIndepOn.notMem_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {i : ι} [Nontrivial R] (hv : LinearIndepOn R v s) (hi : i ∈ s) : v i ∉ Submodule.span R (v '' (s \ {i}))
LinearIndepOn.of_comp 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') (hfv : LinearIndepOn R (⇑f ∘ v) s) : LinearIndepOn R v s
linearIndepOn_iff_notMem_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s : Set ι} : LinearIndepOn K v s ↔ ∀ i ∈ s, v i ∉ Submodule.span K (v '' (s \ {i}))
LinearIndepOn.singleton' 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} {i : ι} (hi : ∀ (r : R), r • v i = 0 → r = 0) : LinearIndepOn R v {i}
LinearIndepOn.eq_zero_of_smul_mem_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {i : ι} (hv : LinearIndepOn R v s) (hi : i ∈ s) (a : R) (ha : a • v i ∈ Submodule.span R (v '' (s \ {i}))) : a = 0
linearIndepOn_finset_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} {s : Finset ι} : LinearIndepOn R v ↑s ↔ ∀ (f : ι → R), ∑ i ∈ s, f i • v i = 0 → ∀ i ∈ s, f i = 0
linearIndepOn_finset_iffₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Finset ι} : LinearIndepOn R v ↑s ↔ ∀ (f g : ι → R), ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i → ∀ i ∈ s, f i = g i
LinearMap.linearIndepOn_iff_of_injOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (f : M →ₗ[R] M') (hf_inj : Set.InjOn ⇑f ↑(Submodule.span R (v '' s))) : LinearIndepOn R (⇑f ∘ v) s ↔ LinearIndepOn R v s
not_linearIndepOn_finset_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} {s : Finset ι} : ¬LinearIndepOn R v ↑s ↔ ∃ f, ∑ i ∈ s, f i • v i = 0 ∧ ∃ i ∈ s, f i ≠ 0
linearIndepOn_iff' 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : LinearIndepOn R v s ↔ ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0
not_linearIndepOn_finset_iffₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Finset ι} : ¬LinearIndepOn R v ↑s ↔ ∃ f g, ∑ i ∈ s, f i • v i = ∑ i ∈ s, g i • v i ∧ ∃ i ∈ s, f i ≠ g i
linearIndepOn_iff'' 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : LinearIndepOn R v s ↔ ∀ (t : Finset ι) (g : ι → R), ↑t ⊆ s → (∀ i ∉ t, g i = 0) → ∑ i ∈ t, g i • v i = 0 → ∀ i ∈ t, g i = 0
linearIndepOn_finset_iffₒₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [LinearOrder R] [CanonicallyOrderedAdd R] [AddRightReflectLE R] [IsCancelAdd M] [DecidableEq ι] {s : Finset ι} : LinearIndepOn R v ↑s ↔ ∀ t ⊆ s, ∀ (f : ι → R), ∑ i ∈ t, f i • v i = ∑ i ∈ s \ t, f i • v i → ∀ i ∈ s, f i = 0
linearIndepOn_iff_disjoint 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : LinearIndepOn R v s ↔ Disjoint (Finsupp.supported R R s) (Finsupp.linearCombination R v).ker
not_linearIndepOn_finset_iffₒₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [LinearOrder R] [CanonicallyOrderedAdd R] [AddRightReflectLE R] [IsCancelAdd M] [DecidableEq ι] {s : Finset ι} : ¬LinearIndepOn R v ↑s ↔ ∃ t ⊆ s, ∃ f, ∑ i ∈ t, f i • v i = ∑ i ∈ s \ t, f i • v i ∧ ∃ i ∈ t, 0 < f i
linearDepOn_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : ¬LinearIndepOn R v s ↔ ∃ f ∈ Finsupp.supported R R s, ∑ i ∈ f.support, f i • v i = 0 ∧ f ≠ 0
linearDepOn_iff' 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : ¬LinearIndepOn R v s ↔ ∃ f ∈ Finsupp.supported R R s, (Finsupp.linearCombination R v) f = 0 ∧ f ≠ 0
linearIndepOn_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : LinearIndepOn R v s ↔ ∀ l ∈ Finsupp.supported R R s, (Finsupp.linearCombination R v) l = 0 → l = 0
linearDepOn_iffₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : ¬LinearIndepOn R v s ↔ ∃ f g, f ∈ Finsupp.supported R R s ∧ g ∈ Finsupp.supported R R s ∧ ∑ i ∈ f.support, f i • v i = ∑ i ∈ g.support, g i • v i ∧ f ≠ g
linearDepOn_iff'ₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : ¬LinearIndepOn R v s ↔ ∃ f g, f ∈ Finsupp.supported R R s ∧ g ∈ Finsupp.supported R R s ∧ (Finsupp.linearCombination R v) f = (Finsupp.linearCombination R v) g ∧ f ≠ g
linearIndepOn_iffₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndepOn R v s ↔ ∀ f ∈ Finsupp.supported R R s, ∀ g ∈ Finsupp.supported R R s, (Finsupp.linearCombination R v) f = (Finsupp.linearCombination R v) g → f = g
linearIndepOn_iff_linearCombinationOnₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] : LinearIndepOn R v s ↔ Function.Injective ⇑(Finsupp.linearCombinationOn ι M R v s)
linearIndepOn_iff_linearCombinationOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Defs
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {v : ι → M} : LinearIndepOn R v s ↔ (Finsupp.linearCombinationOn ι M R v s).ker = ⊥
LinearIndepOn.id_image 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (hs : LinearIndepOn R v s) : LinearIndepOn R id (v '' s)
LinearIndepOn.congr 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {w : ι → M} (hli : LinearIndepOn R v s) (h : Set.EqOn v w s) : LinearIndepOn R w s
linearIndepOn_congr 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {w : ι → M} (h : Set.EqOn v w s) : LinearIndepOn R v s ↔ LinearIndepOn R w s
linearIndepOn_of_finite 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] (s : Set ι) (H : ∀ t ⊆ s, t.Finite → LinearIndepOn R v t) : LinearIndepOn R v s
LinearIndepOn.image_of_comp 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {ι' : Type u_1} {R : Type u_2} {s : Set ι} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (f : ι → ι') (g : ι' → M) (hs : LinearIndepOn R (g ∘ f) s) : LinearIndepOn R g (f '' s)
LinearIndepOn_iff_linearIndepOn_image_injOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] : LinearIndepOn R v s ↔ LinearIndepOn R id (v '' s) ∧ Set.InjOn v s
LinearIndepOn.comp_of_image 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {ι' : Type u_1} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set ι'} {f : ι' → ι} (h : LinearIndepOn R v (f '' s)) (hf : Set.InjOn f s) : LinearIndepOn R (v ∘ f) s
eq_of_linearIndepOn_id_of_span_subtype 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] {s t : Set M} (hs : LinearIndepOn R id s) (h : t ⊆ s) (hst : s ⊆ ↑(Submodule.span R t)) : s = t
linearIndepOn_singleton_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} (R : Type u_2) {M : Type u_4} [Ring R] [IsDomain R] [AddCommGroup M] [Module R M] [Module.IsTorsionFree R M] {v : ι → M} {i : ι} : LinearIndepOn R v {i} ↔ v i ≠ 0
le_of_span_le_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] {s t u : Set M} (hl : LinearIndepOn R id u) (hsu : s ⊆ u) (htu : t ⊆ u) (hst : Submodule.span R s ≤ Submodule.span R t) : s ⊆ t
LinearIndepOn.id_union 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {s t : Set M} (hs : LinearIndepOn R id s) (ht : LinearIndepOn R id t) (hdj : Disjoint (Submodule.span R s) (Submodule.span R t)) : LinearIndepOn R id (s ∪ t)
LinearIndepOn.union 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {t : Set ι} (hs : LinearIndepOn R v s) (ht : LinearIndepOn R v t) (hdj : Disjoint (Submodule.span R (v '' s)) (Submodule.span R (v '' t))) : LinearIndepOn R v (s ∪ t)
LinearIndepOn.id_imageₛ 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] {s : Set M} {f : M →ₗ[R] M'} (hs : LinearIndepOn R id s) (hf_inj : Set.InjOn ⇑f ↑(Submodule.span R s)) : LinearIndepOn R id (⇑f '' s)
LinearIndepOn.map_injOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {M' : Type u_5} {v : ι → M} [Semiring R] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R M'] (hv : LinearIndepOn R v s) (f : M →ₗ[R] M') (hf_inj : Set.InjOn ⇑f ↑(Submodule.span R (v '' s))) : LinearIndepOn R (⇑f ∘ v) s
linearIndepOn_id_union_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {s t : Set M} (hdj : Disjoint s t) : LinearIndepOn R id (s ∪ t) ↔ LinearIndepOn R id s ∧ LinearIndepOn R id t ∧ Disjoint (Submodule.span R s) (Submodule.span R t)
linearIndepOn_union_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{ι : Type u'} {R : Type u_2} {s : Set ι} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {t : Set ι} (hdj : Disjoint s t) : LinearIndepOn R v (s ∪ t) ↔ LinearIndepOn R v s ∧ LinearIndepOn R v t ∧ Disjoint (Submodule.span R (v '' s)) (Submodule.span R (v '' t))
LinearIndepOn.image 📋 Mathlib.LinearAlgebra.LinearIndependent.Basic
{R : Type u_2} {M : Type u_4} {M' : Type u_5} [Ring R] [AddCommGroup M] [AddCommGroup M'] [Module R M] [Module R M'] {s : Set M} {f : M →ₗ[R] M'} (hs : LinearIndepOn R id s) (hf_inj : Disjoint (Submodule.span R s) f.ker) : LinearIndepOn R id (⇑f '' s)
Module.Basis.linearIndepOn 📋 Mathlib.LinearAlgebra.Basis.Basic
{ι : Type u_1} {R : Type u_2} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (b : Module.Basis ι R M) (s : Set ι) : LinearIndepOn R (⇑b) s
nonempty_linearIndependent_set 📋 Mathlib.LinearAlgebra.Dimension.Basic
(R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] [Module R M] : Nonempty { s // LinearIndepOn R id s }
Module.exists_set_linearIndependent_of_lt_rank 📋 Mathlib.LinearAlgebra.Dimension.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {c : Cardinal.{v}} (h : c < Module.rank R M) : ∃ s, Cardinal.mk ↑s = c ∧ LinearIndepOn R id s
Module.exists_set_linearIndependent_of_lt_lift_rank 📋 Mathlib.LinearAlgebra.Dimension.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {c : Cardinal.{w}} (h : Cardinal.lift.{v, w} c < Cardinal.lift.{w, v} (Module.rank R M)) : ∃ s, Cardinal.lift.{w, v} (Cardinal.mk ↑s) = Cardinal.lift.{v, w} c ∧ LinearIndepOn R id s
Module.le_rank_iff_exists_finset 📋 Mathlib.LinearAlgebra.Dimension.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] {n : ℕ} : ↑n ≤ Module.rank R M ↔ ∃ s, s.card = n ∧ LinearIndepOn R id ↑s
LinearIndepOn.encard_le_toENat_rank 📋 Mathlib.LinearAlgebra.Dimension.Basic
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [Nontrivial R] {ι : Type u_1} {v : ι → M} {s : Set ι} (hs : LinearIndepOn R v s) : s.encard ≤ Cardinal.toENat (Module.rank R M)
Module.rank_def 📋 Mathlib.LinearAlgebra.Dimension.Basic
(R : Type u_1) (M : Type u_2) [Semiring R] [AddCommMonoid M] [Module R M] : Module.rank R M = ⨆ ι, Cardinal.mk ↑↑ι
LinearIndepOn.extend 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : Set ι
exists_maximal_linearIndepOn' 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} (R : Type u_2) {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] (v : ι → M) : ∃ s, LinearIndepOn R v s ∧ ∀ (t : Set ι), s ⊆ t → LinearIndepOn R v t → s = t
linearIndepOn_iUnion_of_directed 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {η : Type u_6} {s : η → Set ι} (hs : Directed (fun x1 x2 => x1 ⊆ x2) s) (h : ∀ (i : η), LinearIndepOn R v (s i)) : LinearIndepOn R v (⋃ i, s i)
LinearIndepOn.extend_subset 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : hs.extend hst ⊆ t
LinearIndepOn.subset_extend 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : s ⊆ hs.extend hst
linearIndepOn_sUnion_of_directed 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {s : Set (Set ι)} (hs : DirectedOn (fun x1 x2 => x1 ⊆ x2) s) (h : ∀ a ∈ s, LinearIndepOn R v a) : LinearIndepOn R v (⋃₀ s)
LinearIndepOn.linearIndepOn_extend 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : LinearIndepOn K v (hs.extend hst)
linearIndepOn_biUnion_of_directed 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Semiring R] [AddCommMonoid M] [Module R M] {η : Type u_6} {s : Set η} {t : η → Set ι} (hs : DirectedOn (t ⁻¹'o fun x1 x2 => x1 ⊆ x2) s) (h : ∀ a ∈ s, LinearIndepOn R v (t a)) : LinearIndepOn R v (⋃ a ∈ s, t a)
LinearIndepOn.span_extend_eq_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) : Submodule.span K (hs.extend hst) = Submodule.span K t
LinearIndepOn.subset_span_extend 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) : t ⊆ ↑(Submodule.span K (hs.extend hst))
LinearIndepOn.span_image_extend_eq_span_image 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : Submodule.span K (v '' hs.extend hst) = Submodule.span K (v '' t)
exists_finite_card_le_of_finite_of_linearIndependent_of_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (ht : t.Finite) (hs : LinearIndepOn K id s) (hst : s ⊆ ↑(Submodule.span K t)) : ∃ (h : s.Finite), h.toFinset.card ≤ ht.toFinset.card
LinearIndepOn.image_subset_span_image_extend 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : v '' t ⊆ ↑(Submodule.span K (v '' hs.extend hst))
LinearIndepOn.id_insert 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {x : V} (hs : LinearIndepOn K id s) (hx : x ∉ Submodule.span K s) : LinearIndepOn K id (insert x s)
LinearIndepOn.insert 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s : Set ι} {x : ι} (hs : LinearIndepOn K v s) (hx : v x ∉ Submodule.span K (v '' s)) : LinearIndepOn K v (insert x s)
LinearIndepOn.mem_span_iff_id 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {a : V} (h : LinearIndepOn K id s) : a ∈ Submodule.span K s ↔ LinearIndepOn K id (insert a s) → a ∈ s
linearIndepOn_id_insert_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {a : V} {s : Set V} : LinearIndepOn K id (insert a s) ↔ LinearIndepOn K id s ∧ (a ∈ Submodule.span K s → a ∈ s)
linearIndepOn_id_insert 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {x : V} (hxs : x ∉ s) : LinearIndepOn K id (insert x s) ↔ LinearIndepOn K id s ∧ x ∉ Submodule.span K s
LinearIndepOn.notMem_span_iff_id 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {a : V} (h : LinearIndepOn K id s) : a ∉ Submodule.span K s ↔ LinearIndepOn K id (insert a s) ∧ a ∉ s
exists_linearIndepOn_id_extension 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) : ∃ b ⊆ t, s ⊆ b ∧ t ⊆ ↑(Submodule.span K b) ∧ LinearIndepOn K id b
LinearIndepOn.mem_span_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set ι} {a : ι} {f : ι → V} (h : LinearIndepOn K f s) : f a ∈ Submodule.span K (f '' s) ↔ LinearIndepOn K f (insert a s) → a ∈ s
linearIndepOn_insert_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set ι} {a : ι} {f : ι → V} : LinearIndepOn K f (insert a s) ↔ LinearIndepOn K f s ∧ (f a ∈ Submodule.span K (f '' s) → a ∈ s)
linearIndepOn_insert 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set ι} {a : ι} {f : ι → V} (has : a ∉ s) : LinearIndepOn K f (insert a s) ↔ LinearIndepOn K f s ∧ f a ∉ Submodule.span K (f '' s)
LinearIndepOn.notMem_span_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set ι} {a : ι} {f : ι → V} (h : LinearIndepOn K f s) : f a ∉ Submodule.span K (f '' s) ↔ LinearIndepOn K f (insert a s) ∧ a ∉ s
exists_linearIndepOn_extension 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {v : ι → V} {s t : Set ι} (hs : LinearIndepOn K v s) (hst : s ⊆ t) : ∃ b ⊆ t, s ⊆ b ∧ v '' t ⊆ ↑(Submodule.span K (v '' b)) ∧ LinearIndepOn K v b
linearIndepOn_id_pair 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {x y : V} (hx : x ≠ 0) (hy : ∀ (a : K), a • x ≠ y) : LinearIndepOn K id {x, y}
exists_of_linearIndepOn_of_finite_span 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} {t : Finset V} (hs : LinearIndepOn K id s) (hst : s ⊆ ↑(Submodule.span K ↑t)) : ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = t.card
linearIndepOn_pair_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {K : Type u_3} {V : Type u} [DivisionRing K] [AddCommGroup V] [Module K V] {i j : ι} (v : ι → V) (hij : i ≠ j) (hi : v i ≠ 0) : LinearIndepOn K v {i, j} ↔ ∀ (c : K), c • v i ≠ v j
LinearIndepOn.id_insert' 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {s : Set M} {x : M} (hs : LinearIndepOn R id s) (hx : ∀ (r : R), r • x ∈ Submodule.span R s → r = 0) : LinearIndepOn R id (insert x s)
exists_maximal_linearIndepOn 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} (R : Type u_2) {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] (v : ι → M) : ∃ s, LinearIndepOn R v s ∧ ∀ i ∉ s, ∃ a, a ≠ 0 ∧ a • v i ∈ Submodule.span R (v '' s)
LinearIndepOn.insert' 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} {v : ι → M} [Ring R] [AddCommGroup M] [Module R M] {s : Set ι} {i : ι} (hs : LinearIndepOn R v s) (hx : ∀ (r : R), r • v i ∈ Submodule.span R (v '' s) → r = 0) : LinearIndepOn R v (insert i s)
LinearIndepOn.pair_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {i j : ι} (f : ι → M) (hij : i ≠ j) : LinearIndepOn R f {i, j} ↔ ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0
linearIndepOn_id_iUnion_finite 📋 Mathlib.LinearAlgebra.LinearIndependent.Lemmas
{ι : Type u'} {R : Type u_2} {M : Type u_4} [Ring R] [AddCommGroup M] [Module R M] {f : ι → Set M} (hl : ∀ (i : ι), LinearIndepOn R id (f i)) (hd : ∀ (i : ι) (t : Set ι), t.Finite → i ∉ t → Disjoint (Submodule.span R (f i)) (⨆ i ∈ t, Submodule.span R (f i))) : LinearIndepOn R id (⋃ i, f i)
rank_span_set 📋 Mathlib.LinearAlgebra.Dimension.StrongRankCondition
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [StrongRankCondition R] {s : Set M} (hs : LinearIndepOn R id s) : Module.rank R ↥(Submodule.span R s) = Cardinal.mk ↑s
toENat_rank_span_set 📋 Mathlib.LinearAlgebra.Dimension.StrongRankCondition
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {ι : Type w} [StrongRankCondition R] {v : ι → M} {s : Set ι} (hs : LinearIndepOn R v s) : Cardinal.toENat (Module.rank R ↥(Submodule.span R (v '' s))) = s.encard
Module.instFintypeElemExtendOfFiniteSubtypeMemSubmoduleSpan 📋 Mathlib.LinearAlgebra.Dimension.StrongRankCondition
{R : Type u_2} {M : Type u_3} [DivisionRing R] [AddCommGroup M] [Module R M] {s t : Set M} [Module.Finite R ↥(Submodule.span R t)] (hs : LinearIndepOn R id s) (hst : s ⊆ t) : Fintype ↑(hs.extend hst)
Submodule.exists_finset_span_eq_linearIndepOn 📋 Mathlib.LinearAlgebra.Dimension.StrongRankCondition
(K : Type u_2) {M : Type u_3} [DivisionRing K] [AddCommGroup M] [Module K M] (s : Set M) [Module.Finite K ↥(Submodule.span K s)] : ∃ t, ↑t ⊆ s ∧ t.card = Module.finrank K ↥(Submodule.span K s) ∧ Submodule.span K ↑t = Submodule.span K s ∧ LinearIndepOn K id ↑t
finrank_span_set_eq_card 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [StrongRankCondition R] {s : Set M} [Fintype ↑s] (hs : LinearIndepOn R id s) : Module.finrank R ↥(Submodule.span R s) = s.toFinset.card
finrank_span_finset_eq_card 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] [StrongRankCondition R] {s : Finset M} (hs : LinearIndepOn R id ↑s) : Module.finrank R ↥(Submodule.span R ↑s) = s.card
LinearIndepOn.union_id_of_quotient 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M] {M' : Submodule R M} {s : Set M} (hs : s ⊆ ↑M') (hs' : LinearIndepOn R id s) {t : Set M} (ht : LinearIndepOn R (⇑M'.mkQ) t) : LinearIndepOn R id (s ∪ t)
LinearIndepOn.union_of_quotient 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} {ι : Type w} [Ring R] [AddCommGroup M] [Module R M] {s t : Set ι} {f : ι → M} (hs : LinearIndepOn R f s) (ht : LinearIndepOn R (⇑(Submodule.span R (f '' s)).mkQ ∘ f) t) : LinearIndepOn R f (s ∪ t)
LinearIndepOn.quotient_iff_union 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} {ι : Type w} [Ring R] [AddCommGroup M] [Module R M] {s t : Set ι} {f : ι → M} (hs : LinearIndepOn R f s) (hst : Disjoint s t) : LinearIndepOn R (⇑(Submodule.span R (f '' s)).mkQ ∘ f) t ↔ LinearIndepOn R f (s ∪ t)
linearIndepOn_union_iff_quotient 📋 Mathlib.LinearAlgebra.Dimension.Constructions
{R : Type u} {M : Type v} {ι : Type w} [Ring R] [AddCommGroup M] [Module R M] {s t : Set ι} {f : ι → M} (hst : Disjoint s t) : LinearIndepOn R f (s ∪ t) ↔ LinearIndepOn R f s ∧ LinearIndepOn R (⇑(Submodule.span R (f '' s)).mkQ ∘ f) t
exists_set_linearIndependent_of_lt_rank 📋 Mathlib.LinearAlgebra.Dimension.Finite
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {c : Cardinal.{v}} (h : c < Module.rank R M) : ∃ s, Cardinal.mk ↑s = c ∧ LinearIndepOn R id s
exists_finset_linearIndependent_of_le_rank 📋 Mathlib.LinearAlgebra.Dimension.Finite
{R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ} (hn : ↑n ≤ Module.rank R M) : ∃ s, s.card = n ∧ LinearIndepOn R id ↑s
Module.Basis.subset_extend 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndepOn K id s) : s ⊆ hs.extend ⋯
Module.Basis.extend 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndepOn K id s) : Module.Basis (↑(hs.extend ⋯)) K V
Module.Basis.extendLe 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) : Module.Basis (↑(hs.extend hst)) K V
Module.Basis.range_extend 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndepOn K id s) : Set.range ⇑(Module.Basis.extend hs) = hs.extend ⋯
Module.Basis.coe_extend 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndepOn K id s) : ⇑(Module.Basis.extend hs) = Subtype.val
Module.Basis.extend_apply_self 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s : Set V} (hs : LinearIndepOn K id s) (x : ↑(hs.extend ⋯)) : (Module.Basis.extend hs) x = ↑x
Module.Basis.extendLe_subset 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) : Set.range ⇑(Module.Basis.extendLe hs hst ht) ⊆ t
Module.Basis.subset_extendLe 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) : s ⊆ Set.range ⇑(Module.Basis.extendLe hs hst ht)
Module.Basis.range_extendLe 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) : Set.range ⇑(Module.Basis.extendLe hs hst ht) = hs.extend hst
Module.Basis.coe_extendLe 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) : ⇑(Module.Basis.extendLe hs hst ht) = Subtype.val
Module.Basis.extendLe_apply_self 📋 Mathlib.LinearAlgebra.Basis.VectorSpace
{K : Type u_3} {V : Type u_4} [DivisionRing K] [AddCommGroup V] [Module K V] {s t : Set V} (hs : LinearIndepOn K id s) (hst : s ⊆ t) (ht : ⊤ ≤ Submodule.span K t) (x : ↑(hs.extend hst)) : (Module.Basis.extendLe hs hst ht) x = ↑x
le_rank_iff_exists_linearIndependent 📋 Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
{K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V] [Module.Free K V] {c : Cardinal.{v}} : c ≤ Module.rank K V ↔ ∃ s, Cardinal.mk ↑s = c ∧ LinearIndepOn K id s
Polynomial.natDegree_sum_eq_of_linearIndepOn 📋 Mathlib.Algebra.Polynomial.BigOperators
{R : Type u} {ι : Type w} (s : Finset ι) [CommRing R] {A : Type u_1} [CommRing A] [Algebra R A] {f : ι → Polynomial R} {v : ι → A} (h : LinearIndepOn R v ↑s) : (∑ i ∈ s, v i • Polynomial.map (algebraMap R A) (f i)).natDegree = s.sup fun i => (f i).natDegree
Polynomial.degree_sum_eq_of_linearIndepOn 📋 Mathlib.Algebra.Polynomial.BigOperators
{R : Type u} {ι : Type w} (s : Finset ι) [CommRing R] {A : Type u_1} [CommRing A] [Algebra R A] {f : ι → Polynomial R} {v : ι → A} (h : LinearIndepOn R v ↑s) : (∑ i ∈ s, v i • Polynomial.map (algebraMap R A) (f i)).degree = s.sup fun i => (f i).degree
TensorProduct.LinearIndepOn.tmul_of_flat_left 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u_1} [CommSemiring R] {M : Type u_2} {N : Type u_3} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {ι : Type u_6} {κ : Type u_7} {v : ι → M} {w : κ → N} {s : Set ι} {t : Set κ} [Module.Flat R M] (hv : LinearIndepOn R v s) (hw : LinearIndepOn R w t) : LinearIndepOn R (fun i => v i.1 ⊗ₜ[R] w i.2) (s ×ˢ t)
TensorProduct.LinearIndepOn.tmul_of_flat_right 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u_1} [CommSemiring R] {M : Type u_2} {N : Type u_3} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {ι : Type u_6} {κ : Type u_7} {v : ι → M} {w : κ → N} {s : Set ι} {t : Set κ} [Module.Flat R N] (hv : LinearIndepOn R v s) (hw : LinearIndepOn R w t) : LinearIndepOn R (fun i => v i.1 ⊗ₜ[R] w i.2) (s ×ˢ t)
HasRankNullity.exists_set_linearIndependent 📋 Mathlib.LinearAlgebra.Dimension.RankNullity
{R : Type v} {inst : Ring R} [self : HasRankNullity.{u, v} R] (M : Type u) [AddCommGroup M] [Module R M] : ∃ s, Cardinal.mk ↑s = Module.rank R M ∧ LinearIndepOn R id s
exists_linearIndepOn_of_lt_rank 📋 Mathlib.LinearAlgebra.Dimension.RankNullity
{R : Type u_1} {M : Type u} [Ring R] [AddCommGroup M] [Module R M] [HasRankNullity.{u, u_1} R] [StrongRankCondition R] {s : Set M} (hs : LinearIndepOn R id s) : ∃ t, s ⊆ t ∧ Cardinal.mk ↑t = Module.rank R M ∧ LinearIndepOn R id t
HasRankNullity.mk 📋 Mathlib.LinearAlgebra.Dimension.RankNullity
{R : Type v} [inst : Ring R] (exists_set_linearIndependent : ∀ (M : Type u) [inst_1 : AddCommGroup M] [inst_2 : Module R M], ∃ s, Cardinal.mk ↑s = Module.rank R M ∧ LinearIndepOn R id s) (rank_quotient_add_rank : ∀ {M : Type u} [inst_1 : AddCommGroup M] [inst_2 : Module R M] (N : Submodule R M), Module.rank R (M ⧸ N) + Module.rank R ↥N = Module.rank R M) : HasRankNullity.{u, v} R
Module.exists_dual_forall_apply_eq_one 📋 Mathlib.LinearAlgebra.Dual.Lemmas
{ι : Type u_1} {K : Type u_2} {V : Type u_3} [Field K] [AddCommGroup V] [Module K V] {s : Set ι} {v : ι → V} (hli : LinearIndepOn K v s) : ∃ f, ∀ i ∈ s, f (v i) = 1
LinearIndepOn.id_image_algebraMap_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Algebra
{R : Type u_1} {S : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] [FaithfulSMul S A] {s : Set S} : LinearIndepOn R id (⇑(algebraMap S A) '' s) ↔ LinearIndepOn R id s
LinearIndepOn.algebraMap_comp_iff 📋 Mathlib.LinearAlgebra.LinearIndependent.Algebra
{R : Type u_1} {S : Type u_2} {A : Type u_3} [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R S] [Algebra S A] [Algebra R A] [IsScalarTower R S A] [FaithfulSMul S A] {ι : Type u_4} {v : ι → S} {s : Set ι} : LinearIndepOn R (⇑(algebraMap S A) ∘ v) s ↔ LinearIndepOn R v s
exists_set_linearIndependent_of_isDomain 📋 Mathlib.LinearAlgebra.Dimension.Localization
(R : Type uR) (M : Type uM) [CommRing R] [AddCommGroup M] [Module R M] [IsDomain R] : ∃ s, Cardinal.mk ↑s = Module.rank R M ∧ LinearIndepOn R id s
IsLocalization.linearIndepOn_finsetIntegerMultiple 📋 Mathlib.LinearAlgebra.Dimension.Localization
{R : Type uR} {S : Type uS} [CommRing R] [CommRing S] [Algebra R S] {A : Type u_1} [CommRing A] [Algebra S A] [Algebra R A] [IsScalarTower R S A] (M : Submonoid S) [IsLocalization M A] [FaithfulSMul S A] {s : Finset A} (hs : LinearIndepOn R id ↑s) [DecidableEq S] : LinearIndepOn R id ↑(IsLocalization.finsetIntegerMultiple M s)
Submodule.spanRank_span_of_linearIndepOn 📋 Mathlib.Algebra.Module.SpanRank
{R : Type u_1} {M : Type u} [Semiring R] [AddCommMonoid M] [Module R M] [RankCondition R] (s : Set M) (hs : LinearIndepOn R id s) : (Submodule.span R s).spanRank = Cardinal.mk ↑s
FixedPoints.linearIndependent_smul_of_linearIndependent 📋 Mathlib.FieldTheory.Fixed
(G : Type u) [Group G] (F : Type v) [Field F] [MulSemiringAction G F] {s : Finset F} : LinearIndepOn (↥(FixedPoints.subfield G F)) id ↑s → LinearIndepOn F ⇑(MulAction.toFun G F) ↑s
RootPairing.linearIndepOn_coroot_iff 📋 Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Fintype ι] [AddCommGroup M] [AddCommGroup N] [Field R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsAnisotropic] [P.IsRootSystem] [NeZero 2] {s : Set ι} : LinearIndepOn R (⇑P.coroot) s ↔ LinearIndepOn R (⇑P.root) s
RootPairing.Base.linearIndepOn_coroot 📋 Mathlib.LinearAlgebra.RootSystem.Base
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (self : P.Base) : LinearIndepOn R ⇑P.coroot ↑self.support
RootPairing.Base.linearIndepOn_root 📋 Mathlib.LinearAlgebra.RootSystem.Base
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (self : P.Base) : LinearIndepOn R ⇑P.root ↑self.support
RootPairing.Base.mk 📋 Mathlib.LinearAlgebra.RootSystem.Base
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {P : RootPairing ι R M N} (support : Finset ι) (linearIndepOn_root : LinearIndepOn R ⇑P.root ↑support) (linearIndepOn_coroot : LinearIndepOn R ⇑P.coroot ↑support) (root_mem_or_neg_mem : ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' ↑support) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' ↑support)) (coroot_mem_or_neg_mem : ∀ (i : ι), P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑support) ∨ -P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' ↑support)) : P.Base
RootPairing.linearIndepOn_root_baseOf 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] (f : M →+ ℚ) (hf : ∀ (i : ι), f (P.root i) ≠ 0) : LinearIndepOn R (⇑P.root) (IsAddIndecomposable.baseOf (⇑P.root) f)
RootPairing.ncard_eq_finrank_of_linearIndepOn_of 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [CommRing R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [Nontrivial R] {s : Set ι} (hli : LinearIndepOn R (⇑P.root) s) (hsp : ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) : s.ncard = Module.finrank R M
RootPairing.Base.mk' 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] [P.IsReduced] (s : Set ι) (hli : LinearIndepOn R (⇑P.root) s) (hsp : ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) : P.Base
RootPairing.eq_baseOf_of_linearIndepOn_of_mem_or_neg_mem_closure 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] (s : Set ι) (hli : LinearIndepOn R (⇑P.root) s) (hsp : ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) (f : M →+ ℚ) (hf : ∀ i ∈ s, f (P.root i) = 1) : s = IsAddIndecomposable.baseOf (⇑P.root) f
RootPairing.eq_baseOf_iff 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] (s : Set ι) (f : M →+ ℚ) (hf : ∀ i ∈ s, f (P.root i) = 1) (hf' : ∀ (i : ι), f (P.root i) ≠ 0) : s = IsAddIndecomposable.baseOf (⇑P.root) f ↔ LinearIndepOn R (⇑P.root) s ∧ ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)
RootPairing.coroot_mem_or_neg_mem_closure_of_root 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [Field R] [CharZero R] [Module R M] [Module R N] (P : RootPairing ι R M N) [P.IsRootSystem] [P.IsCrystallographic] [P.IsReduced] (s : Set ι) (hli : LinearIndepOn R (⇑P.root) s) (hsp : ∀ (i : ι), P.root i ∈ AddSubmonoid.closure (⇑P.root '' s) ∨ -P.root i ∈ AddSubmonoid.closure (⇑P.root '' s)) (i : ι) : P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' s) ∨ -P.coroot i ∈ AddSubmonoid.closure (⇑P.coroot '' s)
RootPairing.linearIndepOn_root_baseOf' 📋 Mathlib.LinearAlgebra.RootSystem.BaseExists
{ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [Finite ι] [AddCommGroup M] [AddCommGroup N] [CommRing R] [Module R M] [Module R N] (P : RootPairing ι R M N) [IsDomain R] {S : Type u_6} [LinearOrder S] [CommRing S] [IsStrictOrderedRing S] [Algebra S R] [FaithfulSMul S R] [Module S M] [IsScalarTower S R M] [Module S N] [IsScalarTower S R N] [P.IsValuedIn S] [P.IsCrystallographic] (f : Module.Dual S M) (hf : ∀ (i : ι), f (P.root i) ≠ 0) : LinearIndepOn S (⇑P.root) (IsAddIndecomposable.baseOf ⇑P.root ↑f)
LieAlgebra.Basis.linearIndepOn_root_baseSupp 📋 Mathlib.Algebra.Lie.Basis.Base
{K : Type u_1} {L : Type u_2} [Field K] [CharZero K] [LieRing L] [LieAlgebra K L] [FiniteDimensional K L] {H : LieSubalgebra K L} {ι : Type u_3} [Fintype ι] (b : LieAlgebra.Basis ι H) [LieModule.IsTriangularizable K (↥H) L] [LieAlgebra.IsKilling K L] : LinearIndepOn K (⇑(LieAlgebra.IsKilling.rootSystem H).root) (Set.range b.baseSupp')
LinearMap.le_rank_iff_exists_linearIndependent 📋 Mathlib.LinearAlgebra.Dimension.LinearMap
{K : Type u} {V : Type v} {V' : Type v'} [DivisionRing K] [AddCommGroup V] [Module K V] [AddCommGroup V'] [Module K V'] {c : Cardinal.{v'}} {f : V →ₗ[K] V'} : c ≤ f.rank ↔ ∃ s, Cardinal.lift.{v', v} (Cardinal.mk ↑s) = Cardinal.lift.{v, v'} c ∧ LinearIndepOn K (⇑f) s
exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow_of_essFiniteType 📋 Mathlib.FieldTheory.SeparablyGenerated
{k : Type u_1} {K : Type u_2} [Field k] [Field K] [Algebra k K] (p : ℕ) (hp : Nat.Prime p) (H : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x => x ^ p) ↑s) [ExpChar k p] [Algebra.EssFiniteType k K] : ∃ s, IsTranscendenceBasis k Subtype.val ∧ Algebra.IsSeparable (↥(IntermediateField.adjoin k ↑s)) K
exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow 📋 Mathlib.FieldTheory.SeparablyGenerated
{k : Type u_1} {K : Type u_2} {ι : Type u_3} [Field k] [Field K] [Algebra k K] (p : ℕ) (hp : Nat.Prime p) (H : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x => x ^ p) ↑s) {a : ι → K} (n : ι) [ExpChar k p] (ha' : IsTranscendenceBasis k fun i => a ↑i) : ∃ i, (IsTranscendenceBasis k fun j => a ↑j) ∧ IsSeparable (↥(IntermediateField.adjoin k (a '' {i}ᶜ))) (a i)
exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow' 📋 Mathlib.FieldTheory.SeparablyGenerated
{k : Type u_1} {K : Type u_2} {ι : Type u_3} [Field k] [Field K] [Algebra k K] (p : ℕ) (hp : Nat.Prime p) (H : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x => x ^ p) ↑s) {a : ι → K} [ExpChar k p] (s : Set ι) (n : ι) (ha : IsTranscendenceBasis k fun i => a ↑i) (hn : n ∉ s) : ∃ i, (IsTranscendenceBasis k fun j => a ↑j) ∧ IsSeparable (↥(IntermediateField.adjoin k (a '' (insert n s \ {i})))) (a i)
exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow_of_adjoin_eq_top 📋 Mathlib.FieldTheory.SeparablyGenerated
{k : Type u_1} {K : Type u_2} {ι : Type u_3} [Field k] [Field K] [Algebra k K] (p : ℕ) (hp : Nat.Prime p) (H : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x => x ^ p) ↑s) {a : ι → K} (n : ι) [ExpChar k p] (ha : IntermediateField.adjoin k (Set.range a) = ⊤) (ha' : IsTranscendenceBasis k fun i => a ↑i) : ∃ i, (IsTranscendenceBasis k fun j => a ↑j) ∧ Algebra.IsSeparable (↥(IntermediateField.adjoin k (a '' {i}ᶜ))) K
MvPolynomial.exists_mem_support_not_dvd_of_forall_totalDegree_le 📋 Mathlib.FieldTheory.SeparablyGenerated
{k : Type u_1} {K : Type u_2} {ι : Type u_3} [Field k] [Field K] [Algebra k K] (p : ℕ) (hp : Nat.Prime p) (H : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x => x ^ p) ↑s) {a : ι → K} {F : MvPolynomial ι k} (HF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (MvPolynomial.aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree) (hF0 : F ≠ 0) (hFa : (MvPolynomial.aeval a) F = 0) : ∃ i, ∃ σ ∈ F.support, ¬p ∣ σ i
PointedCone.IsSimplicial.hull 📋 Mathlib.Geometry.Convex.Cone.Simplicial
{R : Type u_1} {M : Type u_2} [Semiring R] [PartialOrder R] [IsOrderedRing R] [AddCommMonoid M] [Module R M] {s : Set M} (hs : s.Finite) (hli : LinearIndepOn R id s) : (PointedCone.hull R s).IsSimplicial
Projectivization.linearIndepOn_pair 📋 Mathlib.LinearAlgebra.Projectivization.Basic
{K : Type u_1} {V : Type u_2} [DivisionRing K] [AddCommGroup V] [Module K V] (D D' : Projectivization K V) : LinearIndepOn K id {D.rep, D'.rep}
isProperLinearSet_iff 📋 Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs
{M : Type u_1} [AddCommMonoid M] {s : Set M} : IsProperLinearSet s ↔ ∃ a t, LinearIndepOn ℕ id ↑t ∧ s = a +ᵥ ↑(AddSubmonoid.closure ↑t)
LinearIndepOn.tmul_of_isDomain 📋 Mathlib.RingTheory.Flat.Domain
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommRing R] [IsDomain R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N] {ι : Type u_6} {κ : Type u_7} {v : ι → M} {w : κ → N} {s : Set ι} {t : Set κ} (hv : LinearIndepOn R v s) (hw : LinearIndepOn R w t) : LinearIndepOn R (fun i => v i.1 ⊗ₜ[R] w i.2) (s ×ˢ t)
linearIndepOn_isGroupLikeElem 📋 Mathlib.RingTheory.Coalgebra.GroupLike
{R : Type u_2} {A : Type u_3} [CommRing R] [IsDomain R] [AddCommGroup A] [Module R A] [Coalgebra R A] [Module.IsTorsionFree R A] : LinearIndepOn R id {a | IsGroupLikeElem R a}

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c