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Theorem ghmid 19211
Description: A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghmid.y 𝑌 = (0g𝑆)
ghmid.z 0 = (0g𝑇)
Assertion
Ref Expression
ghmid (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )

Proof of Theorem ghmid
StepHypRef Expression
1 ghmgrp1 19207 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 eqid 2725 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
3 ghmid.y . . . . . . 7 𝑌 = (0g𝑆)
42, 3grpidcl 18955 . . . . . 6 (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆))
51, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆))
6 eqid 2725 . . . . . 6 (+g𝑆) = (+g𝑆)
7 eqid 2725 . . . . . 6 (+g𝑇) = (+g𝑇)
82, 6, 7ghmlin 19210 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
95, 5, 8mpd3an23 1459 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
102, 6, 3grplid 18957 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g𝑆)𝑌) = 𝑌)
111, 5, 10syl2anc 582 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g𝑆)𝑌) = 𝑌)
1211fveq2d 6904 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = (𝐹𝑌))
139, 12eqtr3d 2767 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌))
14 ghmgrp2 19208 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
15 eqid 2725 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
162, 15ghmf 19209 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1716, 5ffvelcdmd 7098 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) ∈ (Base‘𝑇))
18 ghmid.z . . . . 5 0 = (0g𝑇)
1915, 7, 18grpid 18965 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑌) ∈ (Base‘𝑇)) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2014, 17, 19syl2anc 582 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2113, 20mpbid 231 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹𝑌))
2221eqcomd 2731 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7423  Basecbs 17208  +gcplusg 17261  0gc0g 17449  Grpcgrp 18923   GrpHom cghm 19201
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 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  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 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 8002  df-2nd 8003  df-map 8856  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-ghm 19202
This theorem is referenced by:  ghminv  19212  ghmmhm  19215  ghmpreima  19227  f1ghm0to0  19234  kerf1ghm  19236  ghmqusker  19276  lactghmga  19398  nrhmzr  20514  zrinitorngc  20615  imadrhmcl  20725  srng0  20780  islmhm2  20963  zrh0  21495  chrrhm  21517  zndvds0  21540  ip0l  21624  evlslem2  22086  evlslem3  22087  evlslem6  22088  rhmmpl  22366  rhmply1vr1  22370  0mat2pmat  22721  nmolb2d  24718  nmoi  24728  nmoix  24729  nmoleub  24731  nmoleub2lem2  25126  nmhmcn  25130  dchrptlem2  27286  psgnid  32952  dimkerim  33494  ricdrng1  41948  rhmpsr  41966
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