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Theorem ghomid 21945
 Description: A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ghomid.1 GId
ghomid.2 GId
Assertion
Ref Expression
ghomid GrpOpHom

Proof of Theorem ghomid
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . . . . . 7
2 ghomid.1 . . . . . . 7 GId
31, 2grpoidcl 21797 . . . . . 6
433ad2ant1 978 . . . . 5 GrpOpHom
54, 4jca 519 . . . 4 GrpOpHom
61ghomlin 21944 . . . 4 GrpOpHom
75, 6mpdan 650 . . 3 GrpOpHom
81, 2grpolid 21799 . . . . . 6
93, 8mpdan 650 . . . . 5
109fveq2d 5724 . . . 4
11103ad2ant1 978 . . 3 GrpOpHom
127, 11eqtrd 2467 . 2 GrpOpHom
13 eqid 2435 . . . . . . 7
141, 13elghom 21943 . . . . . 6 GrpOpHom
1514biimp3a 1283 . . . . 5 GrpOpHom
1615simpld 446 . . . 4 GrpOpHom
1716, 4ffvelrnd 5863 . . 3 GrpOpHom
18 ghomid.2 . . . . . 6 GId
1913, 18grpoid 21803 . . . . 5
2019ex 424 . . . 4
21203ad2ant2 979 . . 3 GrpOpHom
2217, 21mpd 15 . 2 GrpOpHom
2312, 22mpbird 224 1 GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697   crn 4871  wf 5442  cfv 5446  (class class class)co 6073  cgr 21766  GIdcgi 21767   GrpOpHom cghom 21937 This theorem is referenced by:  ghomf1olem  25097  grpokerinj  26551  rngohom0  26579 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-grpo 21771  df-gid 21772  df-ghom 21938
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