Theorems · Theorem · group theory
Submonoid.exists_mulEquiv_eq_mgraph
∀ {H : Type u_2} {I : Type u_3} [inst : Monoid H] [inst_1 : Monoid I] {G : Submonoid (H × I)},
Function.Bijective (Prod.fst ∘ ⇑G.subtype) →
Function.Bijective (Prod.snd ∘ ⇑G.subtype) → ∃ e, G = e.toMonoidHom.mgraphGoursat's lemma for monoid isomorphisms.
Let G ≤ H × I be a submonoid of a product of monoids. Assume that the natural maps from G to
both factors are bijective. Then G is the graph of some isomorphism f : H ≃* I.
- Defined in
- Mathlib.Algebra.Group.Graph
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 23 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Monoidstatement and proof · cited by 3,887
- MonoidHomstatement · cited by 3,629
- Submonoidstatement and proof · cited by 3,086
- MulEquivstatement and proof · cited by 1,142
- Function.Bijectivestatement and proof · cited by 863
- MonoidHomClass.toMonoidHomproof · cited by 294
- MulEquiv.toMonoidHomstatement · cited by 126
- Function.Bijective.injectiveproof · cited by 115
- Function.Bijective.surjectiveproof · cited by 114
- Submonoid.subtypestatement and proof · cited by 26
- MonoidHom.mgraphstatement and proof · cited by 8
Cited by1
Results whose statement or proof uses this declaration.
- Subgroup.exists_mulEquiv_eq_graphproof · cited by 0