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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  |-  U  =  (GId `  G )
ghomidOLD.2  |-  T  =  (GId `  H )
Assertion
Ref Expression
ghomidOLD  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  =  T )

Proof of Theorem ghomidOLD
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2423 . . . . . . 7  |-  ran  G  =  ran  G
2 ghomidOLD.1 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidcl 25937 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
433ad2ant1 1027 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  U  e.  ran  G )
54, 4jca 535 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( U  e. 
ran  G  /\  U  e. 
ran  G ) )
61ghomlinOLD 26084 . . . 4  |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
)  /\  ( U  e.  ran  G  /\  U  e.  ran  G ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  ( U G U ) ) )
75, 6mpdan 673 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  ( U G U ) ) )
81, 2grpolid 25939 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
93, 8mpdan 673 . . . . 5  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
109fveq2d 5883 . . . 4  |-  ( G  e.  GrpOp  ->  ( F `  ( U G U ) )  =  ( F `  U ) )
11103ad2ant1 1027 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  ( U G U ) )  =  ( F `
 U ) )
127, 11eqtrd 2464 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
13 eqid 2423 . . . . . . 7  |-  ran  H  =  ran  H
141, 13elghomOLD 26083 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp )  ->  ( F  e.  ( G GrpOpHom  H )  <->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) ) )
1514biimp3a 1365 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F : ran  G --> ran  H  /\  A. x  e.  ran  G A. y  e.  ran  G ( ( F `  x ) H ( F `  y ) )  =  ( F `
 ( x G y ) ) ) )
1615simpld 461 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  H )
1716, 4ffvelrnd 6036 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  e.  ran  H )
18 ghomidOLD.2 . . . . . 6  |-  T  =  (GId `  H )
1913, 18grpoid 25943 . . . . 5  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  ran  H )  -> 
( ( F `  U )  =  T  <-> 
( ( F `  U ) H ( F `  U ) )  =  ( F `
 U ) ) )
2019ex 436 . . . 4  |-  ( H  e.  GrpOp  ->  ( ( F `  U )  e.  ran  H  ->  (
( F `  U
)  =  T  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) ) )
21203ad2ant2 1028 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U )  e. 
ran  H  ->  ( ( F `  U )  =  T  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) ) )
2217, 21mpd 15 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U )  =  T  <->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) ) )
2312, 22mpbird 236 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  =  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   A.wral 2776   ran crn 4852   -->wf 5595   ` cfv 5599  (class class class)co 6303   GrpOpcgr 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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