MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghmmhmb Structured version   Visualization version   GIF version

Theorem ghmmhmb 17872
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 17871 . . 3 (𝑓 ∈ (𝑆 GrpHom 𝑇) → 𝑓 ∈ (𝑆 MndHom 𝑇))
2 eqid 2760 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2760 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
4 eqid 2760 . . . . 5 (+g𝑆) = (+g𝑆)
5 eqid 2760 . . . . 5 (+g𝑇) = (+g𝑇)
6 simpll 807 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Grp)
7 simplr 809 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑇 ∈ Grp)
82, 3mhmf 17541 . . . . . 6 (𝑓 ∈ (𝑆 MndHom 𝑇) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
98adantl 473 . . . . 5 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓:(Base‘𝑆)⟶(Base‘𝑇))
102, 4, 5mhmlin 17543 . . . . . . 7 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
11103expb 1114 . . . . . 6 ((𝑓 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝑓‘(𝑥(+g𝑆)𝑦)) = ((𝑓𝑥)(+g𝑇)(𝑓𝑦)))
1211adantll 752 . . . . 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 17870 . . . 4 (((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ 𝑓 ∈ (𝑆 MndHom 𝑇)) → 𝑓 ∈ (𝑆 GrpHom 𝑇))
1413ex 449 . . 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 2758 1 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143   MndHom cmhm 17534  Grpcgrp 17623   GrpHom cghm 17858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-grp 17626  df-ghm 17859
This theorem is referenced by:  0ghm  17875  resghm2  17878  resghm2b  17879  ghmco  17881  pwsdiagghm  17889  ghmpropd  17899  pwsco1rhm  18940  pwsco2rhm  18941  dchrghm  25180  c0ghm  42421  c0snghm  42426
  Copyright terms: Public domain W3C validator