Loogle!

Result

Found 5087 declarations mentioning Prod.fst. Of these, 158 have a name containing ".fst_".

• Prod.fst_swap 📋 Init.Data.Prod
  {α : Type u_1} {β : Type u_2} {p : α × β} : p.swap.1 = p.2
• Lean.Omega.Prod.fst_mk 📋 Init.Omega.Int
  {α✝ : Type u_1} {x : α✝} {β✝ : Type u_2} {y : β✝} : (x, y).1 = x
• Array.fst_unzip 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array (α × β)} : xs.unzip.1 = Array.map Prod.fst xs
• BitVec.iunfoldr.fst_eq 📋 Init.Data.BitVec.Folds
  {w : ℕ} {α : Type u_1} {f : Fin w → α → α × Bool} (state : ℕ → α) (s : α) (init : s = state 0) (ind : ∀ (i : Fin w), (f i (state ↑i)).1 = state (↑i + 1)) : (BitVec.iunfoldr f s).1 = state w
• List.fst_mem_of_mem_zipIdx 📋 Init.Data.List.Nat.Range
  {α : Type u_1} {x : α × ℕ} {l : List α} {k : ℕ} (h : x ∈ l.zipIdx k) : x.1 ∈ l
• List.fst_eq_of_mem_zipIdx 📋 Init.Data.List.Nat.Range
  {α : Type u_1} {x : α × ℕ} {l : List α} {k : ℕ} (h : x ∈ l.zipIdx k) : x.1 = l[x.2 - k]
• Array.fst_mem_of_mem_zipIdx 📋 Init.Data.Array.Range
  {α : Type u_1} {x : α × ℕ} {xs : Array α} {k : ℕ} (h : x ∈ xs.zipIdx k) : x.1 ∈ xs
• Array.fst_eq_of_mem_zipIdx 📋 Init.Data.Array.Range
  {α : Type u_1} {x : α × ℕ} {xs : Array α} {k : ℕ} (h : x ∈ xs.zipIdx k) : x.1 = xs[x.2 - k]
• Vector.fst_mem_of_mem_zipIdx 📋 Init.Data.Vector.Range
  {α : Type u_1} {n : ℕ} {x : α × ℕ} {xs : Vector α n} {k : ℕ} (h : x ∈ xs.zipIdx k) : x.1 ∈ xs
• Vector.fst_eq_of_mem_zipIdx 📋 Init.Data.Vector.Range
  {α : Type u_1} {n : ℕ} {x : α × ℕ} {xs : Vector α n} {k : ℕ} (h : x ∈ xs.zipIdx k) : x.1 = xs[x.2 - k]
• Std.Internal.List.Prod.fst_comp_toSigma 📋 Std.Data.Internal.List.Associative
  {α : Type u} {β : Type v} : Sigma.fst ∘ Std.Internal.List.Prod.toSigma = Prod.fst
• Std.Do.ExceptConds.fst_false 📋 Std.Do.PostCond
  {ε : Type u} {ps : Std.Do.PostShape} : Std.Do.ExceptConds.false.1 = fun _ε => ⌜False⌝
• Std.Do.ExceptConds.fst_true 📋 Std.Do.PostCond
  {ε : Type u} {ps : Std.Do.PostShape} : Std.Do.ExceptConds.true.1 = fun _ε => ⌜True⌝
• Std.Do.ExceptConds.fst_const 📋 Std.Do.PostCond
  {ε : Type u} {ps : Std.Do.PostShape} (p : Prop) : (Std.Do.ExceptConds.const p).1 = fun _ε => ⌜p⌝
• Std.Do.ExceptConds.fst_and 📋 Std.Do.PostCond
  {ps : Std.Do.PostShape} {ε : Type u} {x₁ x₂ : Std.Do.ExceptConds (Std.Do.PostShape.except ε ps)} : (x₁ ∧ₑ x₂).1 = fun e => spred(x₁.1 e ∧ x₂.1 e)
• Std.Do.ExceptConds.fst_imp 📋 Std.Do.PostCond
  {ps : Std.Do.PostShape} {ε : Type u} {x₁ x₂ : Std.Do.ExceptConds (Std.Do.PostShape.except ε ps)} : (x₁ →ₑ x₂).1 = fun e => spred(x₁.1 e → x₂.1 e)
• Std.Stream.fst_take_zero 📋 Batteries.Data.Stream
  {σ : Type u_1} {α : Type u_2} [Std.Stream σ α] (s : σ) : (Std.Stream.take s 0).1 = []
• Std.Stream.fst_takeTR 📋 Batteries.Data.Stream
  {σ : Type u_1} {α : Type u_2} [Std.Stream σ α] (s : σ) (n : ℕ) : (Std.Stream.takeTR s n).1 = (Std.Stream.take s n).1
• Std.Stream.fst_takeTR_loop 📋 Batteries.Data.Stream
  {σ : Type u_1} {α : Type u_2} [Std.Stream σ α] (s : σ) (acc : List α) (n : ℕ) : (Std.Stream.takeTR.loop s acc n).1 = acc.reverseAux (Std.Stream.take s n).1
• Std.Stream.fst_take_succ 📋 Batteries.Data.Stream
  {σ : Type u_1} {α : Type u_2} {n : ℕ} [Std.Stream σ α] (s : σ) : (Std.Stream.take s (n + 1)).1 = match Std.Stream.next? s with | none => [] | some (a, s) => a :: (Std.Stream.take s n).1
• Prod.fst_injective 📋 Mathlib.Data.Prod.Basic
  {α : Type u_1} {β : Type u_2} [Subsingleton β] : Function.Injective Prod.fst
• Prod.fst_surjective 📋 Mathlib.Data.Prod.Basic
  {α : Type u_1} {β : Type u_2} [h : Nonempty β] : Function.Surjective Prod.fst
• Prod.fst_comp_mk 📋 Mathlib.Data.Prod.Basic
  {α : Type u_1} {β : Type u_2} (x : α) : Prod.fst ∘ Prod.mk x = Function.const β x
• Prod.fst_eq_iff 📋 Mathlib.Data.Prod.Basic
  {α : Type u_1} {β : Type u_2} {p : α × β} {x : α} : p.1 = x ↔ p = (x, p.2)
• Prod.fst_bot 📋 Mathlib.Order.BoundedOrder.Basic
  (α : Type u) (β : Type v) [Bot α] [Bot β] : ⊥.1 = ⊥
• Prod.fst_top 📋 Mathlib.Order.BoundedOrder.Basic
  (α : Type u) (β : Type v) [Top α] [Top β] : ⊤.1 = ⊤
• Prod.fst_inf 📋 Mathlib.Order.Lattice
  (α : Type u) (β : Type v) [Min α] [Min β] (p q : α × β) : (p ⊓ q).1 = p.1 ⊓ q.1
• Prod.fst_sup 📋 Mathlib.Order.Lattice
  (α : Type u) (β : Type v) [Max α] [Max β] (p q : α × β) : (p ⊔ q).1 = p.1 ⊔ q.1
• Prod.fst_toSigma 📋 Mathlib.Data.Sigma.Basic
  {α : Type u_7} {β : Type u_8} (x : α × β) : x.toSigma.fst = x.1
• Prod.fst_comp_toSigma 📋 Mathlib.Data.Sigma.Basic
  {α : Type u_7} {β : Type u_8} : Sigma.fst ∘ Prod.toSigma = Prod.fst
• Equiv.Perm.fst_prodExtendRight 📋 Mathlib.Logic.Equiv.Prod
  {α₁ : Type u_9} {β₁ : Type u_10} [DecidableEq α₁] (a : α₁) (e : Equiv.Perm β₁) (ab : α₁ × β₁) : ((Equiv.Perm.prodExtendRight a e) ab).1 = ab.1
• Set.fst_injOn_graph 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} {s : Set α} {f : α → β} : Set.InjOn Prod.fst (Set.graphOn f s)
• Set.fst_image_prod_subset 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} (s : Set α) (t : Set β) : Prod.fst '' s ×ˢ t ⊆ s
• Set.fst_image_prod 📋 Mathlib.Data.Set.Prod
  {α : Type u_1} {β : Type u_2} (s : Set β) {t : Set α} (ht : t.Nonempty) : Prod.fst '' s ×ˢ t = s
• Prod.fst_inv 📋 Mathlib.Algebra.Notation.Prod
  {G : Type u_8} {H : Type u_9} [Inv G] [Inv H] (p : G × H) : p⁻¹.1 = p.1⁻¹
• Prod.fst_neg 📋 Mathlib.Algebra.Notation.Prod
  {G : Type u_8} {H : Type u_9} [Neg G] [Neg H] (p : G × H) : (-p).1 = -p.1
• Prod.fst_one 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_3} {N : Type u_4} [One M] [One N] : 1.1 = 1
• Prod.fst_star 📋 Mathlib.Algebra.Notation.Prod
  {R : Type u_6} {S : Type u_7} [Star R] [Star S] (x : R × S) : (star x).1 = star x.1
• Prod.fst_zero 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_3} {N : Type u_4} [Zero M] [Zero N] : 0.1 = 0
• Prod.fst_add 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Add M] [Add N] (p q : M × N) : (p + q).1 = p.1 + q.1
• Prod.fst_div 📋 Mathlib.Algebra.Notation.Prod
  {G : Type u_8} {H : Type u_9} [Div G] [Div H] (a b : G × H) : (a / b).1 = a.1 / b.1
• Prod.fst_mul 📋 Mathlib.Algebra.Notation.Prod
  {M : Type u_8} {N : Type u_9} [Mul M] [Mul N] (p q : M × N) : (p * q).1 = p.1 * q.1
• Prod.fst_sub 📋 Mathlib.Algebra.Notation.Prod
  {G : Type u_8} {H : Type u_9} [Sub G] [Sub H] (a b : G × H) : (a - b).1 = a.1 - b.1
• Prod.fst_add_snd 📋 Mathlib.Algebra.Group.Prod
  {M : Type u_3} {N : Type u_4} [AddZeroClass M] [AddZeroClass N] (p : M × N) : (p.1, 0) + (0, p.2) = p
• Prod.fst_mul_snd 📋 Mathlib.Algebra.Group.Prod
  {M : Type u_3} {N : Type u_4} [MulOneClass M] [MulOneClass N] (p : M × N) : (p.1, 1) * (1, p.2) = p
• OrderHom.fst_coe 📋 Mathlib.Order.Hom.Basic
  {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (self : α × β) : OrderHom.fst self = self.1
• Prod.fst_iInf 📋 Mathlib.Order.CompleteLattice.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).1 = ⨅ i, (f i).1
• Prod.fst_iSup 📋 Mathlib.Order.CompleteLattice.Basic
  {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).1 = ⨆ i, (f i).1
• Prod.fst_sInf 📋 Mathlib.Order.CompleteLattice.Basic
  {α : Type u_1} {β : Type u_2} [InfSet α] [InfSet β] (s : Set (α × β)) : (sInf s).1 = sInf (Prod.fst '' s)
• Prod.fst_sSup 📋 Mathlib.Order.CompleteLattice.Basic
  {α : Type u_1} {β : Type u_2} [SupSet α] [SupSet β] (s : Set (α × β)) : (sSup s).1 = sSup (Prod.fst '' s)
• Prod.fst_vsub 📋 Mathlib.Algebra.AddTorsor.Basic
  {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (p₁ p₂ : P × P') : (p₁ -ᵥ p₂).1 = p₁.1 -ᵥ p₂.1
• Prod.fst_vadd 📋 Mathlib.Algebra.AddTorsor.Basic
  {G : Type u_1} {G' : Type u_2} {P : Type u_3} {P' : Type u_4} [AddGroup G] [AddGroup G'] [AddTorsor G P] [AddTorsor G' P'] (v : G × G') (p : P × P') : (v +ᵥ p).1 = v.1 +ᵥ p.1
• Multiset.fst_prod 📋 Mathlib.Algebra.BigOperators.Group.Multiset.Basic
  {M : Type u_5} {N : Type u_6} [CommMonoid M] [CommMonoid N] (s : Multiset (M × N)) : s.prod.1 = (Multiset.map Prod.fst s).prod
• Multiset.fst_sum 📋 Mathlib.Algebra.BigOperators.Group.Multiset.Basic
  {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] (s : Multiset (M × N)) : s.sum.1 = (Multiset.map Prod.fst s).sum
• WCovBy.fst_ofLex 📋 Mathlib.Data.Prod.Lex
  {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : Lex (α × β)} (h : a ⩿ b) : (ofLex a).1 ⩿ (ofLex b).1
• LatticeHom.fst_apply 📋 Mathlib.Order.Hom.Lattice
  {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (x : α × β) : LatticeHom.fst x = x.1
• Prod.fst_eq_or_snd_eq_of_wcovBy 📋 Mathlib.Order.Cover
  {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β} : x ⩿ y → x.1 = y.1 ∨ x.2 = y.2
• AddActionHom.fst_apply 📋 Mathlib.GroupTheory.GroupAction.Hom
  (M : Type u_1) (α : Type u_2) (β : Type u_3) [VAdd M α] [VAdd M β] : ⇑(AddActionHom.fst M α β) = Prod.fst
• MulActionHom.fst_apply 📋 Mathlib.GroupTheory.GroupAction.Hom
  (M : Type u_1) (α : Type u_2) (β : Type u_3) [SMul M α] [SMul M β] : ⇑(MulActionHom.fst M α β) = Prod.fst
• Prod.fst_natCast 📋 Mathlib.Data.Nat.Cast.Prod
  {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) : (↑n).1 = ↑n
• Prod.fst_ofNat 📋 Mathlib.Data.Nat.Cast.Prod
  {α : Type u_1} {β : Type u_2} [AddMonoidWithOne α] [AddMonoidWithOne β] (n : ℕ) [n.AtLeastTwo] : (OfNat.ofNat n).1 = OfNat.ofNat n
• Prod.fst_intCast 📋 Mathlib.Data.Int.Cast.Prod
  {α : Type u_1} {β : Type u_2} [AddGroupWithOne α] [AddGroupWithOne β] (n : ℤ) : (↑n).1 = ↑n
• Prod.fst_prod 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_1} {M : Type u_3} {N : Type u_4} [CommMonoid M] [CommMonoid N] {s : Finset ι} {f : ι → M × N} : (∏ c ∈ s, f c).1 = ∏ c ∈ s, (f c).1
• Prod.fst_sum 📋 Mathlib.Algebra.BigOperators.Pi
  {ι : Type u_1} {M : Type u_3} {N : Type u_4} [AddCommMonoid M] [AddCommMonoid N] {s : Finset ι} {f : ι → M × N} : (∑ c ∈ s, f c).1 = ∑ c ∈ s, (f c).1
• Finsupp.fst_sumFinsuppEquivProdFinsupp 📋 Mathlib.Data.Finsupp.Basic
  {α : Type u_12} {β : Type u_13} {γ : Type u_14} [Zero γ] (f : α ⊕ β →₀ γ) (x : α) : (Finsupp.sumFinsuppEquivProdFinsupp f).1 x = f (Sum.inl x)
• Finsupp.fst_sumFinsuppAddEquivProdFinsupp 📋 Mathlib.Data.Finsupp.Basic
  {M : Type u_5} [AddMonoid M] {α : Type u_12} {β : Type u_13} (f : α ⊕ β →₀ M) (x : α) : (Finsupp.sumFinsuppAddEquivProdFinsupp f).1 x = f (Sum.inl x)
• AlgHom.fst_apply 📋 Mathlib.Algebra.Algebra.Prod
  (R : Type u_1) {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A × B) : (AlgHom.fst R A B) a = a.1
• LinearMap.fst_apply 📋 Mathlib.LinearAlgebra.Prod
  {R : Type u} {M : Type v} {M₂ : Type w} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] (x : M × M₂) : (LinearMap.fst R M M₂) x = x.1
• Finset.antidiagonal.fst_le 📋 Mathlib.Algebra.Order.Antidiag.Prod
  {A : Type u_1} [AddCommMonoid A] [PartialOrder A] [CanonicallyOrderedAdd A] [Finset.HasAntidiagonal A] {n : A} {kl : A × A} (hlk : kl ∈ Finset.antidiagonal n) : kl.1 ≤ n
• Finset.Nat.antidiagonal.fst_lt 📋 Mathlib.Data.Finset.NatAntidiagonal
  {n : ℕ} {kl : ℕ × ℕ} (hlk : kl ∈ Finset.antidiagonal n) : kl.1 < n + 1
• LinearMap.fst_prodOfFinsuppNat 📋 Mathlib.LinearAlgebra.Finsupp.Pi
  {R : Type u_1} {M : Type u_2} {P : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P] (f : P × M →ₗ[R] M) (x : ℕ →₀ P) : (f.prodOfFinsuppNat x).1 = x 0
• Finsupp.fst_sumFinsuppLEquivProdFinsupp 📋 Mathlib.LinearAlgebra.Finsupp.SumProd
  {M : Type u_2} (R : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] {α : Type u_7} {β : Type u_8} (f : α ⊕ β →₀ M) (x : α) : ((Finsupp.sumFinsuppLEquivProdFinsupp R) f).1 x = f (Sum.inl x)
• NonUnitalAlgHom.fst_toFun 📋 Mathlib.Algebra.Algebra.NonUnitalHom
  (R : Type u) [Monoid R] (A : Type v) (B : Type w) [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] (self : A × B) : (NonUnitalAlgHom.fst R A B) self = self.1
• Prod.fst_kstar 📋 Mathlib.Algebra.Order.Kleene
  {α : Type u_1} {β : Type u_2} [KleeneAlgebra α] [KleeneAlgebra β] (a : α × β) : (KStar.kstar a).1 = KStar.kstar a.1
• StarAlgHom.fst_apply 📋 Mathlib.Algebra.Star.StarAlgHom
  (R : Type u_1) (A : Type u_2) (B : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] (self : A × B) : (StarAlgHom.fst R A B) self = self.1
• NonUnitalStarAlgHom.fst_apply 📋 Mathlib.Algebra.Star.StarAlgHom
  (R : Type u_1) (A : Type u_2) (B : Type u_3) [Monoid R] [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B] (self : A × B) : (NonUnitalStarAlgHom.fst R A B) self = self.1
• Unitization.fst_inl 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_1} (A : Type u_2) [Zero A] (r : R) : (Unitization.inl r).toProd.1 = r
• Unitization.fst_inr 📋 Mathlib.Algebra.Algebra.Unitization
  (R : Type u_1) {A : Type u_2} [Zero R] (a : A) : (↑a).toProd.1 = 0
• Unitization.fst_one 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_1} {A : Type u_2} [One R] [Zero A] : (Unitization.toProd 1).1 = 1
• Unitization.fst_zero 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_3} {A : Type u_4} [Zero R] [Zero A] : (Unitization.toProd 0).1 = 0
• Unitization.fst_neg 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_3} {A : Type u_4} [Neg R] [Neg A] (x : Unitization R A) : (-x).toProd.1 = -x.toProd.1
• Unitization.fst_star 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_1} {A : Type u_2} [Star R] [Star A] (x : Unitization R A) : (star x).toProd.1 = star x.toProd.1
• Unitization.fst_smul 📋 Mathlib.Algebra.Algebra.Unitization
  {S : Type u_2} {R : Type u_3} {A : Type u_4} [SMul S R] [SMul S A] (s : S) (x : Unitization R A) : (s • x).toProd.1 = s • x.toProd.1
• Unitization.fst_add 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_3} {A : Type u_4} [Add R] [Add A] (x₁ x₂ : Unitization R A) : (x₁ + x₂).toProd.1 = x₁.toProd.1 + x₂.toProd.1
• Unitization.fst_mul 📋 Mathlib.Algebra.Algebra.Unitization
  {R : Type u_1} {A : Type u_2} [Mul R] [Add A] [Mul A] [SMul R A] (x₁ x₂ : Unitization R A) : (x₁ * x₂).toProd.1 = x₁.toProd.1 * x₂.toProd.1
• LinearPMap.fst_apply 📋 Mathlib.LinearAlgebra.LinearPMap
  {R : Type u_1} [Ring R] {E : Type u_4} [AddCommGroup E] [Module R E] {F : Type u_5} [AddCommGroup F] [Module R F] (p : Submodule R E) (p' : Submodule R F) (x : ↥(p.prod p')) : ↑(LinearPMap.fst p p') x = (↑x).1
• AddAction.fst_mem_orbit_of_mem_orbit 📋 Mathlib.GroupTheory.GroupAction.Basic
  {M : Type u} [AddMonoid M] {α : Type v} [AddAction M α] {β : Type u_1} [AddAction M β] {x y : α × β} (h : x ∈ AddAction.orbit M y) : x.1 ∈ AddAction.orbit M y.1
• MulAction.fst_mem_orbit_of_mem_orbit 📋 Mathlib.GroupTheory.GroupAction.Basic
  {M : Type u} [Monoid M] {α : Type v} [MulAction M α] {β : Type u_1} [MulAction M β] {x y : α × β} (h : x ∈ MulAction.orbit M y) : x.1 ∈ MulAction.orbit M y.1
• Nat.fst_mem_divisors_of_mem_antidiagonal 📋 Mathlib.NumberTheory.Divisors
  {n : ℕ} {x : ℕ × ℕ} (h : x ∈ n.divisorsAntidiagonal) : x.1 ∈ n.divisors
• Prod.fst_zmod_cast 📋 Mathlib.Data.ZMod.Basic
  {n : ℕ} {R : Type u_1} [AddGroupWithOne R] {S : Type u_2} [AddGroupWithOne S] (a : ZMod n) : a.cast.1 = a.cast
• Nat.fst_maxPowDvdDiv 📋 Mathlib.Data.Nat.MaxPowDiv
  (p n : ℕ) : (p.maxPowDvdDiv n).1 = padicValNat p n
• CategoryTheory.Prod.fst_obj 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] (X : C × D) : (CategoryTheory.Prod.fst C D).obj X = X.1
• CategoryTheory.Prod.fst_map 📋 Mathlib.CategoryTheory.Products.Basic
  (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (D : Type u₂) [CategoryTheory.Category.{v₂, u₂} D] {X✝ Y✝ : C × D} (f : X✝ ⟶ Y✝) : (CategoryTheory.Prod.fst C D).map f = f.1
• Filter.Tendsto.fst_nhds 📋 Mathlib.Topology.Constructions.SumProd
  {Y : Type v} {Z : Type u_2} [TopologicalSpace Y] [TopologicalSpace Z] {X : Type u_5} {l : Filter X} {f : X → Y × Z} {p : Y × Z} (h : Filter.Tendsto f l (nhds p)) : Filter.Tendsto (fun a => (f a).1) l (nhds p.1)
• ContinuousAddMonoidHom.fst_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  (A : Type u_2) (B : Type u_3) [AddMonoid A] [AddMonoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousAddMonoidHom.fst A B) self = self.1
• ContinuousMonoidHom.fst_toFun 📋 Mathlib.Topology.Algebra.ContinuousMonoidHom
  (A : Type u_2) (B : Type u_3) [Monoid A] [Monoid B] [TopologicalSpace A] [TopologicalSpace B] (self : A × B) : (ContinuousMonoidHom.fst A B) self = self.1
• ContinuousMap.fst_apply 📋 Mathlib.Topology.ContinuousMap.Basic
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] : ⇑ContinuousMap.fst = Prod.fst
• ContinuousLinearEquiv.fst_equivOfRightInverse 📋 Mathlib.Topology.Algebra.Module.Equiv
  {R : Type u_1} [Ring R] {M : Type u_3} [TopologicalSpace M] [AddCommGroup M] [Module R M] {M₂ : Type u_4} [TopologicalSpace M₂] [AddCommGroup M₂] [Module R M₂] [IsTopologicalAddGroup M] (f₁ : M →L[R] M₂) (f₂ : M₂ →L[R] M) (h : Function.RightInverse ⇑f₂ ⇑f₁) (x : M) : ((ContinuousLinearEquiv.equivOfRightInverse f₁ f₂ h) x).1 = f₁ x
• MonoidWithZeroHom.fst_apply_coe 📋 Mathlib.Algebra.GroupWithZero.ProdHom
  {G₀ : Type u_1} {H₀ : Type u_2} [GroupWithZero G₀] [GroupWithZero H₀] (x : G₀ˣ × H₀ˣ) : (MonoidWithZeroHom.fst G₀ H₀) ↑x = ↑x.1
• QuadraticMap.Isometry.fst_apply 📋 Mathlib.LinearAlgebra.QuadraticForm.Prod
  {R : Type u_2} {M₁ : Type u_3} (M₂ : Type u_4) {P : Type u_7} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid P] [Module R M₁] [Module R M₂] [Module R P] (Q₁ : QuadraticMap R M₁ P) (self : M₁ × M₂) : (QuadraticMap.Isometry.fst M₂ Q₁) self = self.1
• LieHom.fst_apply 📋 Mathlib.Algebra.Lie.Prod
  {R : Type u_1} {L₁ : Type u_2} {L₂ : Type u_3} [CommRing R] [LieRing L₁] [LieAlgebra R L₁] [LieRing L₂] [LieAlgebra R L₂] (x : L₁ × L₂) : (LieHom.fst R L₁ L₂) x = x.1
• List.TProd.fst_mk 📋 Mathlib.Data.Prod.TProd
  {ι : Type u} {α : ι → Type v} (i : ι) (l : List ι) (f : (i : ι) → α i) : (List.TProd.mk (i :: l) f).1 = f i
• MeasureTheory.Measure.fst_apply 📋 Mathlib.MeasureTheory.Measure.Prod
  {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {ρ : MeasureTheory.Measure (α × β)} {s : Set α} (hs : MeasurableSet s) : ρ.fst s = ρ (Prod.fst ⁻¹' s)
• AffineMap.fst_lineMap 📋 Mathlib.LinearAlgebra.AffineSpace.AffineMap
  {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [Ring k] [AddCommGroup V1] [Module k V1] [AddTorsor V1 P1] [AddCommGroup V2] [Module k V2] [AddTorsor V2 P2] (p₀ p₁ : P1 × P2) (c : k) : ((AffineMap.lineMap p₀ p₁) c).1 = (AffineMap.lineMap p₀.1 p₁.1) c
• ContinuousAffineMap.fst_decompEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
  (R : Type u_1) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompEquiv R V Q) f).1 = f 0
• ContinuousAffineMap.fst_decompLinearEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
  (R : Type u_1) (S : Type u_2) (V : Type u_3) (W : Type u_4) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [IsTopologicalAddGroup W] (f : V →ᴬ[R] W) : ((ContinuousAffineMap.decompLinearEquiv R S V W) f).1 = f 0
• ContinuousAffineMap.fst_decompAffineEquiv 📋 Mathlib.Topology.Algebra.ContinuousAffineMap
  (R : Type u_1) (S : Type u_2) (V : Type u_3) {W : Type u_4} (Q : Type u_5) [Ring S] [Ring R] [AddCommGroup V] [Module R V] [TopologicalSpace V] [IsTopologicalAddGroup V] [AddCommGroup W] [Module R W] [TopologicalSpace W] [Module S W] [SMulCommClass R S W] [ContinuousConstSMul S W] [AddTorsor W Q] [TopologicalSpace Q] [IsTopologicalAddGroup W] [IsTopologicalAddTorsor Q] (f : V →ᴬ[R] Q) : ((ContinuousAffineMap.decompAffineEquiv R S V Q) f).1 = f 0
• Set.SMulAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [SMul G P] {x y : ↑(s.smulAntidiagonal t a)} [IsCancelSMul G P] : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• Set.VAddAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.SMulAntidiagonal
  {G : Type u_1} {P : Type u_2} {s : Set G} {t : Set P} {a : P} [VAdd G P] {x y : ↑(s.vaddAntidiagonal t a)} [IsCancelVAdd G P] : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• OrderAddMonoidHom.fst_apply 📋 Mathlib.Algebra.Order.Monoid.Lex
  (α : Type u_1) (β : Type u_2) [AddMonoid α] [PartialOrder α] [AddMonoid β] [Preorder β] (self : α × β) : (OrderAddMonoidHom.fst α β) self = self.1
• OrderMonoidHom.fst_apply 📋 Mathlib.Algebra.Order.Monoid.Lex
  (α : Type u_1) (β : Type u_2) [Monoid α] [PartialOrder α] [Monoid β] [Preorder β] (self : α × β) : (OrderMonoidHom.fst α β) self = self.1
• NonemptyInterval.fst_le_snd 📋 Mathlib.Order.Interval.Basic
  {α : Type u_6} [LE α] (self : NonemptyInterval α) : self.toProd.1 ≤ self.toProd.2
• NonemptyInterval.fst_sup 📋 Mathlib.Order.Interval.Basic
  {α : Type u_1} [Lattice α] (s t : NonemptyInterval α) : (s ⊔ t).toProd.1 = s.toProd.1 ⊓ t.toProd.1
• NonemptyInterval.fst_dual 📋 Mathlib.Order.Interval.Basic
  {α : Type u_1} [LE α] (s : NonemptyInterval α) : (NonemptyInterval.dual s).toProd.1 = OrderDual.toDual s.toProd.2
• NonemptyInterval.fst_natCast 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [NatCast α] (n : ℕ) : (↑n).toProd.1 = ↑n
• NonemptyInterval.fst_one 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [One α] : (NonemptyInterval.toProd 1).1 = 1
• NonemptyInterval.fst_zero 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [Zero α] : (NonemptyInterval.toProd 0).1 = 0
• NonemptyInterval.fst_inv 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [CommGroup α] [PartialOrder α] [IsOrderedMonoid α] (s : NonemptyInterval α) : s⁻¹.toProd.1 = s.toProd.2⁻¹
• NonemptyInterval.fst_neg 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [AddCommGroup α] [PartialOrder α] [IsOrderedAddMonoid α] (s : NonemptyInterval α) : (-s).toProd.1 = -s.toProd.2
• NonemptyInterval.fst_add 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : (s + t).toProd.1 = s.toProd.1 + t.toProd.1
• NonemptyInterval.fst_mul 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s t : NonemptyInterval α) : (s * t).toProd.1 = s.toProd.1 * t.toProd.1
• NonemptyInterval.fst_nsmul 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd.1 = n • s.toProd.1
• NonemptyInterval.fst_pow 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd.1 = s.toProd.1 ^ n
• NonemptyInterval.fst_div 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s t : NonemptyInterval α) : (s / t).toProd.1 = s.toProd.1 / t.toProd.2
• NonemptyInterval.fst_sub 📋 Mathlib.Algebra.Order.Interval.Basic
  {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s t : NonemptyInterval α) : (s - t).toProd.1 = s.toProd.1 - t.toProd.2
• Set.AddAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [AddCommMonoid α] [IsCancelAdd α] {s t : Set α} {a : α} {x y : ↑(s.addAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• Set.MulAntidiagonal.fst_eq_fst_iff_snd_eq_snd 📋 Mathlib.Data.Set.MulAntidiagonal
  {α : Type u_1} [CommMonoid α] [IsCancelMul α] {s t : Set α} {a : α} {x y : ↑(s.mulAntidiagonal t a)} : (↑x).1 = (↑y).1 ↔ (↑x).2 = (↑y).2
• AlgebraicGeometry.Scheme.IsLocallyDirected.fst_inv_eq_snd_inv 📋 Mathlib.AlgebraicGeometry.Gluing
  {J : Type w} [CategoryTheory.Category.{v, w} J] (F : CategoryTheory.Functor J AlgebraicGeometry.Scheme) [∀ {i j : J} (f : i ⟶ j), AlgebraicGeometry.IsOpenImmersion (F.map f)] [(F.comp AlgebraicGeometry.Scheme.forget).IsLocallyDirected] [Quiver.IsThin J] {i j : J} (k₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)) {U : (F.obj i).Opens} (h₁ : AlgebraicGeometry.Scheme.Hom.opensRange (F.map k₁.snd.1) ≤ U) (h₂ : AlgebraicGeometry.Scheme.Hom.opensRange (F.map k₂.snd.1) ≤ U) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.fst ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)) (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.isoOpensRange (F.map k₁.snd.1)).inv (F.map k₁.snd.2)) = CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.pullback.snd ((F.obj i).homOfLE h₁) ((F.obj i).homOfLE h₂)) (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.isoOpensRange (F.map k₂.snd.1)).inv (F.map k₂.snd.2))
• Prod.fst_exp 📋 Mathlib.Analysis.Normed.Algebra.Exponential
  {𝔸 : Type u_1} {𝔹 : Type u_2} [NormedRing 𝔸] [NormedAlgebra ℚ 𝔸] [CompleteSpace 𝔸] [NormedRing 𝔹] [NormedAlgebra ℚ 𝔹] [CompleteSpace 𝔹] (x : 𝔸 × 𝔹) : (NormedSpace.exp x).1 = NormedSpace.exp x.1
• Algebra.Generators.CotangentSpace.fst_compEquiv_apply 📋 Mathlib.RingTheory.Kaehler.JacobiZariski
  {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] [Algebra R S] {T : Type u₃} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {ι : Type w₁} {σ : Type w₂} (Q : Algebra.Generators S T ι) (P : Algebra.Generators R S σ) (x : (Q.comp P).toExtension.CotangentSpace) : ((Algebra.Generators.CotangentSpace.compEquiv Q P) x).1 = (Algebra.Extension.CotangentSpace.map (Q.ofComp P).toExtensionHom) x
• Prod.fst_sConvexComb 📋 Mathlib.Geometry.Convex.ConvexSpace.Prod
  {R : Type u_2} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] {X : Type u_3} {Y : Type u_4} [Convexity.ConvexSpace R X] [Convexity.ConvexSpace R Y] (w : Convexity.StdSimplex R (X × Y)) : (Convexity.sConvexComb w).1 = Convexity.iConvexComb w Prod.fst
• Prod.fst_iConvexComb 📋 Mathlib.Geometry.Convex.ConvexSpace.Prod
  {I : Type u_1} {R : Type u_2} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] {X : Type u_3} {Y : Type u_4} [Convexity.ConvexSpace R X] [Convexity.ConvexSpace R Y] (w : Convexity.StdSimplex R I) (f : I → X × Y) : (Convexity.iConvexComb w f).1 = Convexity.iConvexComb w fun i => (f i).1
• Prod.fst_convexCombPair 📋 Mathlib.Geometry.Convex.ConvexSpace.Prod
  {R : Type u_2} [Semiring R] [PartialOrder R] [IsStrictOrderedRing R] {X : Type u_3} {Y : Type u_4} [Convexity.ConvexSpace R X] [Convexity.ConvexSpace R Y] (a b : R) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) (x y : X × Y) : (Convexity.convexCombPair a b ha hb hab x y).1 = Convexity.convexCombPair a b ha hb hab x.1 y.1
• ContinuousAffineMap.fst_decompLinearIsometryEquiv 📋 Mathlib.Analysis.Normed.Affine.ContinuousAffineMap
  (𝕜 : Type u_1) (R : Type u_2) (V : Type u_3) (W : Type u_4) [SeminormedAddCommGroup V] [SeminormedAddCommGroup W] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 V] [NormedSpace 𝕜 W] [Ring R] [Module R W] [ContinuousConstSMul R W] [SMulCommClass 𝕜 R W] (f : V →ᴬ[𝕜] W) : ((ContinuousAffineMap.decompLinearIsometryEquiv 𝕜 R V W) f).1 = f 0
• CategoryTheory.Bicategory.Prod.fst_obj 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] (a✝ : B × C) : (CategoryTheory.Bicategory.Prod.fst B C).obj a✝ = a✝.1
• CategoryTheory.Bicategory.Prod.fst_map 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] {X✝ Y✝ : B × C} (a✝ : X✝ ⟶ Y✝) : (CategoryTheory.Bicategory.Prod.fst B C).map a✝ = a✝.1
• CategoryTheory.Bicategory.Prod.fst_map₂ 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] {a✝ b✝ : B × C} {f✝ g✝ : a✝ ⟶ b✝} (a✝¹ : f✝ ⟶ g✝) : (CategoryTheory.Bicategory.Prod.fst B C).map₂ a✝¹ = a✝¹.1
• CategoryTheory.Bicategory.Prod.fst_mapId_hom 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] (x : B × C) : ((CategoryTheory.Bicategory.Prod.fst B C).mapId x).hom = CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.id x.1)
• CategoryTheory.Bicategory.Prod.fst_mapId_inv 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] (x : B × C) : ((CategoryTheory.Bicategory.Prod.fst B C).mapId x).inv = CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.id x.1)
• CategoryTheory.Bicategory.Prod.fst_mapComp_hom 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] {a✝ b✝ c✝ : B × C} (f : a✝ ⟶ b✝) (g : b✝ ⟶ c✝) : ((CategoryTheory.Bicategory.Prod.fst B C).mapComp f g).hom = CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f.1 g.1)
• CategoryTheory.Bicategory.Prod.fst_mapComp_inv 📋 Mathlib.CategoryTheory.Bicategory.Product
  (B : Type u₁) [CategoryTheory.Bicategory B] (C : Type u₂) [CategoryTheory.Bicategory C] {a✝ b✝ c✝ : B × C} (f : a✝ ⟶ b✝) (g : b✝ ⟶ c✝) : ((CategoryTheory.Bicategory.Prod.fst B C).mapComp f g).inv = CategoryTheory.CategoryStruct.id (CategoryTheory.CategoryStruct.comp f.1 g.1)
• TwoPointing.fst_ne_snd 📋 Mathlib.Data.TwoPointing
  {α : Type u_3} (self : TwoPointing α) : self.toProd.1 ≠ self.toProd.2
• CategoryTheory.FunctorToTypes.prod.fst_app 📋 Mathlib.CategoryTheory.Limits.Shapes.FunctorToTypes
  {C : Type u} [CategoryTheory.Category.{v, u} C] {F G : CategoryTheory.Functor C (Type w)} (x✝ : C) : CategoryTheory.FunctorToTypes.prod.fst.app x✝ = TypeCat.ofHom fun a => a.1
• Finset.addETransformLeft.fst_add_snd_subset 📋 Mathlib.Combinatorics.Additive.ETransform
  {α : Type u_1} [DecidableEq α] [AddGroup α] (e : α) (x : Finset α × Finset α) : (Finset.addETransformLeft e x).1 + (Finset.addETransformLeft e x).2 ⊆ x.1 + x.2
• Finset.addETransformRight.fst_add_snd_subset 📋 Mathlib.Combinatorics.Additive.ETransform
  {α : Type u_1} [DecidableEq α] [AddGroup α] (e : α) (x : Finset α × Finset α) : (Finset.addETransformRight e x).1 + (Finset.addETransformRight e x).2 ⊆ x.1 + x.2
• Finset.mulETransformLeft.fst_mul_snd_subset 📋 Mathlib.Combinatorics.Additive.ETransform
  {α : Type u_1} [DecidableEq α] [Group α] (e : α) (x : Finset α × Finset α) : (Finset.mulETransformLeft e x).1 * (Finset.mulETransformLeft e x).2 ⊆ x.1 * x.2
• Finset.mulETransformRight.fst_mul_snd_subset 📋 Mathlib.Combinatorics.Additive.ETransform
  {α : Type u_1} [DecidableEq α] [Group α] (e : α) (x : Finset α × Finset α) : (Finset.mulETransformRight e x).1 * (Finset.mulETransformRight e x).2 ⊆ x.1 * x.2
• SimpleGraph.Dart.fst_ne_snd 📋 Mathlib.Combinatorics.SimpleGraph.Dart
  {V : Type u_1} {G : SimpleGraph V} (d : G.Dart) : d.toProd.1 ≠ d.toProd.2
• SimpleGraph.Walk.fst_darts_getElem 📋 Mathlib.Combinatorics.SimpleGraph.Walk.Basic
  {V : Type u} {G : SimpleGraph V} {u v : V} {p : G.Walk u v} {i : ℕ} (hi : i < p.darts.length) : p.darts[i].toProd.1 = p.support.dropLast[i]
• AddMonoid.Coprod.fst_toProd 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} [AddMonoid M] [AddMonoid N] (x : AddMonoid.Coprod M N) : (AddMonoid.Coprod.toProd x).1 = AddMonoid.Coprod.fst x
• Monoid.Coprod.fst_toProd 📋 Mathlib.GroupTheory.Coprod.Basic
  {M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] (x : Monoid.Coprod M N) : (Monoid.Coprod.toProd x).1 = Monoid.Coprod.fst x
• ProbabilityTheory.Kernel.fst_eq 📋 Mathlib.Probability.Kernel.Composition.MapComap
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) : κ.fst = κ.map Prod.fst
• ProbabilityTheory.Kernel.fst_apply 📋 Mathlib.Probability.Kernel.Composition.MapComap
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α) : κ.fst a = MeasureTheory.Measure.map Prod.fst (κ a)
• ProbabilityTheory.Kernel.fst_real_apply 📋 Mathlib.Probability.Kernel.Composition.MapComap
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α) {s : Set β} (hs : MeasurableSet s) : (κ.fst a).real s = (κ a).real {p | p.1 ∈ s}
• ProbabilityTheory.Kernel.fst_apply' 📋 Mathlib.Probability.Kernel.Composition.MapComap
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} (κ : ProbabilityTheory.Kernel α (β × γ)) (a : α) {s : Set β} (hs : MeasurableSet s) : (κ.fst a) s = (κ a) {p | p.1 ∈ s}
• LucasLehmer.X.fst_intCast 📋 Mathlib.NumberTheory.LucasLehmer
  {q : ℕ} (n : ℤ) : (↑n).1 = ↑n
• LucasLehmer.X.fst_natCast 📋 Mathlib.NumberTheory.LucasLehmer
  {q : ℕ} (n : ℕ) : (↑n).1 = ↑n
• Ordinal.CNF.fst_le_log 📋 Mathlib.SetTheory.Ordinal.CantorNormalForm
  {b o : Ordinal.{u}} {x : Ordinal.{u} × Ordinal.{u}} : x ∈ Ordinal.CNF b o → x.1 ≤ Ordinal.log b o

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:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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.

5. 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 a114d38 serving mathlib revision 38b5af2