Theorems · Theorem · group theory
MonoidHom.map_mclosure
∀ {M : Type u_1} {N : Type u_2} [inst : MulOneClass M] [inst_1 : MulOneClass N] {F : Type u_4} [inst_2 : FunLike F M N]
[mc : MonoidHomClass F M N] (f : F) (s : Set M), Submonoid.map f (Submonoid.closure s) = Submonoid.closure (⇑f '' s)The image under a MonoidHom of the Submonoid generated by a set equals the Submonoid
generated by the image of the set.
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 65 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 · cited by 62,936
- Setstatement and proof · cited by 53,352
- Set.imagestatement · cited by 5,609
- Submonoidstatement and proof · cited by 3,086
- FunLikestatement and proof · cited by 2,560
- MulOneClassstatement and proof · cited by 1,018
- MonoidHomClassstatement and proof · cited by 244
- Submonoid.mapstatement · cited by 190
- Submonoid.closurestatement · cited by 167
- GaloisInsertion.gcproof · cited by 137
- Submonoid.gc_map_comapproof · cited by 16
- Set.image_preimageproof · cited by 16
Cited by10
Results whose statement or proof uses this declaration.
- Submonoid.map_powersproof · cited by 10
- Monoid.fg_of_surjectiveproof · cited by 4
- Monoid.Coprod.mclosure_range_inl_union_inrproof · cited by 3
- FreeMonoid.mrange_liftproof · cited by 2
- MonoidHom.mclosure_rangeproof · cited by 2
- Submonoid.FG.map_injectiveproof · cited by 1
- AddMonoidAlgebra.mem_closure_of_mem_span_closureproof · cited by 1
- Submonoid.FG.mapproof · cited by 1
- Submonoid.smul_closureproof · cited by 0
- MonoidAlgebra.mem_closure_of_mem_span_closureproof · cited by 0