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Theorem ghomidOLD 26085
 Description: Obsolete version of ghmid 16882 as of 15-Mar-2020. A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ghomidOLD.1 GId
ghomidOLD.2 GId
Assertion
Ref Expression
ghomidOLD GrpOpHom

Proof of Theorem ghomidOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2423 . . . . . . 7
2 ghomidOLD.1 . . . . . . 7 GId
31, 2grpoidcl 25937 . . . . . 6
433ad2ant1 1027 . . . . 5 GrpOpHom
54, 4jca 535 . . . 4 GrpOpHom
61ghomlinOLD 26084 . . . 4 GrpOpHom
75, 6mpdan 673 . . 3 GrpOpHom
81, 2grpolid 25939 . . . . . 6
93, 8mpdan 673 . . . . 5
109fveq2d 5883 . . . 4
11103ad2ant1 1027 . . 3 GrpOpHom
127, 11eqtrd 2464 . 2 GrpOpHom
13 eqid 2423 . . . . . . 7
141, 13elghomOLD 26083 . . . . . 6 GrpOpHom
1514biimp3a 1365 . . . . 5 GrpOpHom
1615simpld 461 . . . 4 GrpOpHom
1716, 4ffvelrnd 6036 . . 3 GrpOpHom
18 ghomidOLD.2 . . . . . 6 GId
1913, 18grpoid 25943 . . . . 5
2019ex 436 . . . 4
21203ad2ant2 1028 . . 3 GrpOpHom
2217, 21mpd 15 . 2 GrpOpHom
2312, 22mpbird 236 1 GrpOpHom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 188   wa 371   w3a 983   wceq 1438   wcel 1869  wral 2776   crn 4852  wf 5595  cfv 5599  (class class class)co 6303  cgr 25906  GIdcgi 25907   GrpOpHom cghomOLD 26077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-grpo 25911  df-gid 25912  df-ghomOLD 26078 This theorem is referenced by:  ghomf1olem  30314  grpokerinj  32141  rngohom0  32169
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