Theorem ghmmhmb 19149

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 19148	. . 3 ⊢ (𝑓 ∈ (𝑆 GrpHom 𝑇) → 𝑓 ∈ (𝑆 MndHom 𝑇))
2		eqid 2726	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2726	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4		eqid 2726	. . . . 5 ⊢ (+g‘𝑆) = (+g‘𝑆)
5		eqid 2726	. . . . 5 ⊢ (+g‘𝑇) = (+g‘𝑇)
6		simpll 764	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Grp)
7		simplr 766	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑇 ∈ Grp)
8	2, 3	mhmf 18716	. . . . . 6 ⊢ (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
9	8	adantl 481	. . . . 5 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
10	2, 4, 5	mhmlin 18720	. . . . . . 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 1117	. . . . . 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 711	. . . . 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 19147	. . . 4 ⊢ (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓 ∈ (𝑆 GrpHom 𝑇))
14	13	ex 412	. . 3 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓 ∈ (𝑆 GrpHom 𝑇)))
15	1, 14	impbid2 225	. 2 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↔ 𝑓 ∈ (𝑆 MndHom 𝑇)))
16	15	eqrdv 2724	1 ⊢ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟶wf 6532  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  +_gcplusg 17203   MndHom cmhm 18708  Grpcgrp 18860   GrpHom cghm 19135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-mhm 18710  df-grp 18863  df-ghm 19136

This theorem is referenced by:  0ghm  19152  resghm2  19155  resghm2b  19156  ghmco  19158  pwsdiagghm  19166  ghmpropd  19178  c0ghm  20360  c0snghm  20363  pwsco1rhm  20401  pwsco2rhm  20402  dchrghm  27139