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Theorem ghmmhmb 19188
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.
StepHypRef Expression
1 ghmmhm 19187 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) → 𝑓 ∈ (𝑆 MndHom 𝑇))
2 eqid 2728 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2728 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
4 eqid 2728 . . . . 5 (+g𝑆) = (+g𝑆)
5 eqid 2728 . . . . 5 (+g𝑇) = (+g𝑇)
6 simpll 765 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Grp)
7 simplr 767 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑇 ∈ Grp)
82, 3mhmf 18753 . . . . . 6 (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
98adantl 480 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
102, 4, 5mhmlin 18757 . . . . . . 7 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
11103expb 1117 . . . . . 6 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
1211adantll 712 . . . . 5 ((((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
132, 3, 4, 5, 6, 7, 9, 12isghmd 19186 . . . 4 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓 ∈ (𝑆 GrpHom 𝑇))
1413ex 411 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓 ∈ (𝑆 GrpHom 𝑇)))
151, 14impbid2 225 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↔ 𝑓 ∈ (𝑆 MndHom 𝑇)))
1615eqrdv 2726 1 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wf 6549  cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240   MndHom cmhm 18745  Grpcgrp 18897   GrpHom cghm 19174
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-grp 18900  df-ghm 19175
This theorem is referenced by:  0ghm  19191  resghm2  19194  resghm2b  19195  ghmco  19197  pwsdiagghm  19205  ghmpropd  19217  c0ghm  20407  c0snghm  20410  pwsco1rhm  20448  pwsco2rhm  20449  dchrghm  27209
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