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Theorem ghmmhmb 17652
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 17651 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) → 𝑓 ∈ (𝑆 MndHom 𝑇))
2 eqid 2620 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2620 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
4 eqid 2620 . . . . 5 (+g𝑆) = (+g𝑆)
5 eqid 2620 . . . . 5 (+g𝑇) = (+g𝑇)
6 simpll 789 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Grp)
7 simplr 791 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑇 ∈ Grp)
82, 3mhmf 17321 . . . . . 6 (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
98adantl 482 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
102, 4, 5mhmlin 17323 . . . . . . 7 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
11103expb 1264 . . . . . 6 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
1211adantll 749 . . . . 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 17650 . . . 4 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓 ∈ (𝑆 GrpHom 𝑇))
1413ex 450 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓 ∈ (𝑆 GrpHom 𝑇)))
151, 14impbid2 216 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑓 ∈ (𝑆 GrpHom 𝑇) ↔ 𝑓 ∈ (𝑆 MndHom 𝑇)))
1615eqrdv 2618 1 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  wf 5872  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922   MndHom cmhm 17314  Grpcgrp 17403   GrpHom cghm 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-mhm 17316  df-grp 17406  df-ghm 17639
This theorem is referenced by:  0ghm  17655  resghm2  17658  resghm2b  17659  ghmco  17661  pwsdiagghm  17669  ghmpropd  17679  pwsco1rhm  18719  pwsco2rhm  18720  dchrghm  24962  c0ghm  41676  c0snghm  41681
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