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Theorem ghmid 19175
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 19171 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 eqid 2728 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
3 ghmid.y . . . . . . 7 𝑌 = (0g𝑆)
42, 3grpidcl 18921 . . . . . 6 (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆))
51, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆))
6 eqid 2728 . . . . . 6 (+g𝑆) = (+g𝑆)
7 eqid 2728 . . . . . 6 (+g𝑇) = (+g𝑇)
82, 6, 7ghmlin 19174 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
95, 5, 8mpd3an23 1460 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
102, 6, 3grplid 18923 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g𝑆)𝑌) = 𝑌)
111, 5, 10syl2anc 583 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g𝑆)𝑌) = 𝑌)
1211fveq2d 6901 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = (𝐹𝑌))
139, 12eqtr3d 2770 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌))
14 ghmgrp2 19172 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
15 eqid 2728 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
162, 15ghmf 19173 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1716, 5ffvelcdmd 7095 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) ∈ (Base‘𝑇))
18 ghmid.z . . . . 5 0 = (0g𝑇)
1915, 7, 18grpid 18931 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑌) ∈ (Base‘𝑇)) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2014, 17, 19syl2anc 583 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2113, 20mpbid 231 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹𝑌))
2221eqcomd 2734 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17179  +gcplusg 17232  0gc0g 17420  Grpcgrp 18889   GrpHom cghm 19166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892  df-ghm 19167
This theorem is referenced by:  ghminv  19176  ghmmhm  19179  ghmpreima  19191  f1ghm0to0  19198  kerf1ghm  19200  ghmqusker  19237  lactghmga  19359  nrhmzr  20473  zrinitorngc  20574  imadrhmcl  20684  srng0  20739  islmhm2  20922  zrh0  21438  chrrhm  21460  zndvds0  21483  ip0l  21567  evlslem2  22024  evlslem3  22025  evlslem6  22026  0mat2pmat  22637  nmolb2d  24634  nmoi  24644  nmoix  24645  nmoleub  24647  nmoleub2lem2  25042  nmhmcn  25046  dchrptlem2  27197  psgnid  32818  dimkerim  33321  ricdrng1  41764  rhmmpl  41786
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