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Theorem ghmmhm 17871
Description: A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Assertion
Ref Expression
ghmmhm (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))

Proof of Theorem ghmmhm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmgrp1 17863 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 grpmnd 17630 . . . 4 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
31, 2syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
4 ghmgrp2 17864 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
5 grpmnd 17630 . . . 4 (𝑇 ∈ Grp → 𝑇 ∈ Mnd)
64, 5syl 17 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
73, 6jca 555 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
8 eqid 2760 . . . 4 (Base‘𝑆) = (Base‘𝑆)
9 eqid 2760 . . . 4 (Base‘𝑇) = (Base‘𝑇)
108, 9ghmf 17865 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11 eqid 2760 . . . . . 6 (+g𝑆) = (+g𝑆)
12 eqid 2760 . . . . . 6 (+g𝑇) = (+g𝑇)
138, 11, 12ghmlin 17866 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
14133expb 1114 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
1514ralrimivva 3109 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
16 eqid 2760 . . . 4 (0g𝑆) = (0g𝑆)
17 eqid 2760 . . . 4 (0g𝑇) = (0g𝑇)
1816, 17ghmid 17867 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
1910, 15, 183jca 1123 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
208, 9, 11, 12, 16, 17ismhm 17538 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
217, 19, 20sylanbrc 701 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wf 6045  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Mndcmnd 17495   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:  ghmmhmb  17872  ghmmulg  17873  resghm2  17878  ghmco  17881  ghmeql  17884  symgtrinv  18092  frgpup3lem  18390  gsummulglem  18541  gsumzinv  18545  gsuminv  18546  gsummulc1  18806  gsummulc2  18807  pwsco2rhm  18941  gsumvsmul  19129  evlslem2  19714  evls1gsumadd  19891  zrhpsgnmhm  20132  mat2pmatmul  20738  pm2mp  20832  cayhamlem4  20895  tsmsinv  22152  plypf1  24167  amgmlem  24915  lgseisenlem4  25302  mendring  38264  amgmwlem  43061  amgmlemALT  43062
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