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Theorem ghomid 21801
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  |-  U  =  (GId `  G )
ghomid.2  |-  T  =  (GId `  H )
Assertion
Ref Expression
ghomid  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  =  T )

Proof of Theorem ghomid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . . . . . 7  |-  ran  G  =  ran  G
2 ghomid.1 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidcl 21653 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
433ad2ant1 978 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  U  e.  ran  G )
54, 4jca 519 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( U  e. 
ran  G  /\  U  e. 
ran  G ) )
61ghomlin 21800 . . . 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 650 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  ( U G U ) ) )
81, 2grpolid 21655 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
93, 8mpdan 650 . . . . 5  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
109fveq2d 5672 . . . 4  |-  ( G  e.  GrpOp  ->  ( F `  ( U G U ) )  =  ( F `  U ) )
11103ad2ant1 978 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  ( U G U ) )  =  ( F `
 U ) )
127, 11eqtrd 2419 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
13 eqid 2387 . . . . . . 7  |-  ran  H  =  ran  H
141, 13elghom 21799 . . . . . 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 1283 . . . . 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 446 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  H )
1716, 4ffvelrnd 5810 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  e.  ran  H )
18 ghomid.2 . . . . . 6  |-  T  =  (GId `  H )
1913, 18grpoid 21659 . . . . 5  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  ran  H )  -> 
( ( F `  U )  =  T  <-> 
( ( F `  U ) H ( F `  U ) )  =  ( F `
 U ) ) )
2019ex 424 . . . 4  |-  ( H  e.  GrpOp  ->  ( ( F `  U )  e.  ran  H  ->  (
( F `  U
)  =  T  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) ) )
21203ad2ant2 979 . . 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 224 1  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  =  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   GrpOpcgr 21622  GIdcgi 21623   GrpOpHom cghom 21793
This theorem is referenced by:  ghomf1olem  24884  grpokerinj  26251  rngohom0  26279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-riota 6485  df-grpo 21627  df-gid 21628  df-ghom 21794
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