Theorem ghmmhmb 19467

Description: Group homomorphisms and monoid homomorphisms coincide. (Thus, GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
ghmmhmb	⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))

Proof of Theorem ghmmhmb

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmmhm 19466	. . 3 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) → 𝑓 ∈ (𝑆 MndHom 𝑇))
2		eqid 2763	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2763	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4		eqid 2763	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
5		eqid 2763	. . . . 5 ⊢ (+g‘𝑇) = (+g‘𝑇)
6		simpll 779	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Grp)
7		simplr 781	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑇 ∈ Grp)
8	2, 3	mhmf 19010	. . . . . 6 ⊢ (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
9	8	adantl 487	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
10	2, 4, 5	mhmlin 19014	. . . . . . 7 ⊢ ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
11	10	3expb 1138	. . . . . 6 ⊢ ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
12	11	adantll 727	. . . . 5 ⊢ ((((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g‘𝑆)𝑦)) = ((𝑓‘𝑥)(+g‘𝑇)(𝑓‘𝑦)))
13	2, 3, 4, 5, 6, 7, 9, 12	isghmd 19465	. . . 4 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓 ∈ (𝑆 GrpHom 𝑇))
14	13	ex 418	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓 ∈ (𝑆 GrpHom 𝑇)))
15	1, 14	impbid2 229	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↔ 𝑓 ∈ (𝑆 MndHom 𝑇)))
16	15	eqrdv 2761	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454   MndHom cmhm 19002  Grpcgrp 19170   GrpHom cghm 19453

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-map 8874  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-grp 19173  df-ghm 19454

This theorem is used by:  0ghm  19470  resghm2  19473  resghm2b  19474  ghmco  19476  pwsdiagghm  19484  ghmpropd  19496  c0ghm  20718  c0snghm  20721  pwsco1rhm  20768  pwsco2rhm  20769  dchrghm  27610