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Theorem ghomid 24031
 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 2454 . . . . . . 7
2 ghomid.1 . . . . . . 7 GId
31, 2grpoidcl 23883 . . . . . 6
433ad2ant1 1009 . . . . 5 GrpOpHom
54, 4jca 532 . . . 4 GrpOpHom
61ghomlin 24030 . . . 4 GrpOpHom
75, 6mpdan 668 . . 3 GrpOpHom
81, 2grpolid 23885 . . . . . 6
93, 8mpdan 668 . . . . 5
109fveq2d 5806 . . . 4
11103ad2ant1 1009 . . 3 GrpOpHom
127, 11eqtrd 2495 . 2 GrpOpHom
13 eqid 2454 . . . . . . 7
141, 13elghom 24029 . . . . . 6 GrpOpHom
1514biimp3a 1319 . . . . 5 GrpOpHom
1615simpld 459 . . . 4 GrpOpHom
1716, 4ffvelrnd 5956 . . 3 GrpOpHom
18 ghomid.2 . . . . . 6 GId
1913, 18grpoid 23889 . . . . 5
2019ex 434 . . . 4
21203ad2ant2 1010 . . 3 GrpOpHom
2217, 21mpd 15 . 2 GrpOpHom
2312, 22mpbird 232 1 GrpOpHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 965   wceq 1370   wcel 1758  wral 2799   crn 4952  wf 5525  cfv 5529  (class class class)co 6203  cgr 23852  GIdcgi 23853   GrpOpHom cghom 24023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-grpo 23857  df-gid 23858  df-ghom 24024 This theorem is referenced by:  ghomf1olem  27480  grpokerinj  28921  rngohom0  28949
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