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Theorem ghmid 19253
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 19249 . . . . . 6 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
2 eqid 2735 . . . . . . 7 (Base‘𝑆) = (Base‘𝑆)
3 ghmid.y . . . . . . 7 𝑌 = (0g𝑆)
42, 3grpidcl 18996 . . . . . 6 (𝑆 ∈ Grp → 𝑌 ∈ (Base‘𝑆))
51, 4syl 17 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑌 ∈ (Base‘𝑆))
6 eqid 2735 . . . . . 6 (+g𝑆) = (+g𝑆)
7 eqid 2735 . . . . . 6 (+g𝑇) = (+g𝑇)
82, 6, 7ghmlin 19252 . . . . 5 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑌 ∈ (Base‘𝑆) ∧ 𝑌 ∈ (Base‘𝑆)) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
95, 5, 8mpd3an23 1462 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = ((𝐹𝑌)(+g𝑇)(𝐹𝑌)))
102, 6, 3grplid 18998 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑌 ∈ (Base‘𝑆)) → (𝑌(+g𝑆)𝑌) = 𝑌)
111, 5, 10syl2anc 584 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑌(+g𝑆)𝑌) = 𝑌)
1211fveq2d 6911 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(𝑌(+g𝑆)𝑌)) = (𝐹𝑌))
139, 12eqtr3d 2777 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → ((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌))
14 ghmgrp2 19250 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
15 eqid 2735 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
162, 15ghmf 19251 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1716, 5ffvelcdmd 7105 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) ∈ (Base‘𝑇))
18 ghmid.z . . . . 5 0 = (0g𝑇)
1915, 7, 18grpid 19006 . . . 4 ((𝑇 ∈ Grp ∧ (𝐹𝑌) ∈ (Base‘𝑇)) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2014, 17, 19syl2anc 584 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (((𝐹𝑌)(+g𝑇)(𝐹𝑌)) = (𝐹𝑌) ↔ 0 = (𝐹𝑌)))
2113, 20mpbid 232 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 0 = (𝐹𝑌))
2221eqcomd 2741 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹𝑌) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ghm 19244
This theorem is referenced by:  ghminv  19254  ghmmhm  19257  ghmpreima  19269  f1ghm0to0  19276  kerf1ghm  19278  ghmqusker  19318  lactghmga  19438  nrhmzr  20554  zrinitorngc  20659  imadrhmcl  20815  srng0  20872  islmhm2  21055  zrh0  21542  chrrhm  21564  zndvds0  21587  ip0l  21672  evlslem2  22121  evlslem3  22122  evlslem6  22123  rhmmpl  22403  rhmply1vr1  22407  0mat2pmat  22758  nmolb2d  24755  nmoi  24765  nmoix  24766  nmoleub  24768  nmoleub2lem2  25163  nmhmcn  25167  dchrptlem2  27324  psgnid  33100  dimkerim  33655  lvecendof1f1o  33661  ricdrng1  42515  rhmpsr  42539
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