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Theorem ghmmhm 19019
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 19011 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
21grpmndd 18761 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Mnd)
3 ghmgrp2 19012 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
43grpmndd 18761 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Mnd)
5 eqid 2737 . . . 4 (Base‘𝑆) = (Base‘𝑆)
6 eqid 2737 . . . 4 (Base‘𝑇) = (Base‘𝑇)
75, 6ghmf 19013 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
8 eqid 2737 . . . . . 6 (+g𝑆) = (+g𝑆)
9 eqid 2737 . . . . . 6 (+g𝑇) = (+g𝑇)
105, 8, 9ghmlin 19014 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
11103expb 1121 . . . 4 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
1211ralrimivva 3198 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
13 eqid 2737 . . . 4 (0g𝑆) = (0g𝑆)
14 eqid 2737 . . . 4 (0g𝑇) = (0g𝑇)
1513, 14ghmid 19015 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
167, 12, 153jca 1129 . 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)))
175, 6, 8, 9, 13, 14ismhm 18604 . 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))))
182, 4, 16, 17syl21anbrc 1345 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹 ∈ (𝑆 MndHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wf 6493  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Mndcmnd 18557   MndHom cmhm 18600   GrpHom cghm 19006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-mhm 18602  df-grp 18752  df-ghm 19007
This theorem is referenced by:  ghmmhmb  19020  ghmmulg  19021  resghm2  19026  ghmco  19029  ghmeql  19032  symgtrinv  19255  frgpup3lem  19560  gsummulglem  19719  gsumzinv  19723  gsuminv  19724  gsummulc1  20031  gsummulc2  20032  pwsco2rhm  20174  gsumvsmul  20389  zrhpsgnmhm  20991  evlslem2  21492  evlsgsumadd  21504  evls1gsumadd  21693  mat2pmatmul  22083  pm2mp  22177  cayhamlem4  22240  tsmsinv  23502  plypf1  25576  amgmlem  26342  lgseisenlem4  26729  gsumvsmul1  31896  rhmpreimaidl  32203  rhmcomulmpl  40743  rhmmpl  40744  selvcllem4  40762  selvadd  40766  selvmul  40767  mendring  41522  amgmwlem  47256  amgmlemALT  47257
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