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Theorem ghmid 19015
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 19011 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 eqid 2737 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
3 ghmid.y . . . . . . 7 𝑌 = (0g𝑆)
42, 3grpidcl 18779 . . . . . 6 (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆))
51, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆))
6 eqid 2737 . . . . . 6 (+g𝑆) = (+g𝑆)
7 eqid 2737 . . . . . 6 (+g𝑇) = (+g𝑇)
82, 6, 7ghmlin 19014 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
95, 5, 8mpd3an23 1464 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
102, 6, 3grplid 18781 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g𝑆)𝑌) = 𝑌)
111, 5, 10syl2anc 585 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g𝑆)𝑌) = 𝑌)
1211fveq2d 6847 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = (𝐹𝑌))
139, 12eqtr3d 2779 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌))
14 ghmgrp2 19012 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
15 eqid 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
162, 15ghmf 19013 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1716, 5ffvelcdmd 7037 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) ∈ (Base‘𝑇))
18 ghmid.z . . . . 5 0 = (0g𝑇)
1915, 7, 18grpid 18787 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑌) ∈ (Base‘𝑇)) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2014, 17, 19syl2anc 585 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2113, 20mpbid 231 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹𝑌))
2221eqcomd 2743 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  cfv 6497  (class class class)co 7358  Basecbs 17084  +gcplusg 17134  0gc0g 17322  Grpcgrp 18749   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-0g 17324  df-mgm 18498  df-sgrp 18547  df-mnd 18558  df-grp 18752  df-ghm 19007
This theorem is referenced by:  ghminv  19016  ghmmhm  19019  ghmpreima  19031  ghmf1  19038  lactghmga  19188  f1ghm0to0  20175  f1rhm0to0ALT  20176  kerf1ghm  20178  srng0  20322  islmhm2  20502  zrh0  20917  chrrhm  20937  zndvds0  20960  ip0l  21043  evlslem2  21492  evlslem3  21493  evlslem6  21494  0mat2pmat  22088  nmolb2d  24085  nmoi  24095  nmoix  24096  nmoleub  24098  nmoleub2lem2  24482  nmhmcn  24486  dchrptlem2  26616  psgnid  31949  ghmqusker  32201  dimkerim  32325  imadrhmcl  40719  ricdrng1  40720  rhmmpl  40744  nrhmzr  46178  zrinitorngc  46305
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