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Theorem ghomid 21032
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 2283 . . . . . . 7  |-  ran  G  =  ran  G
2 ghomid.1 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidcl 20884 . . . . . 6  |-  ( G  e.  GrpOp  ->  U  e.  ran  G )
433ad2ant1 976 . . . . 5  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  U  e.  ran  G )
54, 4jca 518 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( U  e. 
ran  G  /\  U  e. 
ran  G ) )
61ghomlin 21031 . . . 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 649 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  ( U G U ) ) )
81, 2grpolid 20886 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
93, 8mpdan 649 . . . . 5  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
109fveq2d 5529 . . . 4  |-  ( G  e.  GrpOp  ->  ( F `  ( U G U ) )  =  ( F `  U ) )
11103ad2ant1 976 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  ( U G U ) )  =  ( F `
 U ) )
127, 11eqtrd 2315 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
13 eqid 2283 . . . . . . 7  |-  ran  H  =  ran  H
141, 13elghom 21030 . . . . . 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 1281 . . . . 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 445 . . . 4  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  F : ran  G --> ran  H )
17 ffvelrn 5663 . . . 4  |-  ( ( F : ran  G --> ran  H  /\  U  e. 
ran  G )  -> 
( F `  U
)  e.  ran  H
)
1816, 4, 17syl2anc 642 . . 3  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( F `  U )  e.  ran  H )
19 ghomid.2 . . . . . 6  |-  T  =  (GId `  H )
2013, 19grpoid 20890 . . . . 5  |-  ( ( H  e.  GrpOp  /\  ( F `  U )  e.  ran  H )  -> 
( ( F `  U )  =  T  <-> 
( ( F `  U ) H ( F `  U ) )  =  ( F `
 U ) ) )
2120ex 423 . . . 4  |-  ( H  e.  GrpOp  ->  ( ( F `  U )  e.  ran  H  ->  (
( F `  U
)  =  T  <->  ( ( F `  U ) H ( F `  U ) )  =  ( F `  U
) ) ) )
22213ad2ant2 977 . . 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
) ) ) )
2318, 22mpd 14 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U )  =  T  <->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) ) )
2412, 23mpbird 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   GrpOpHom cghom 21024
This theorem is referenced by:  ghomf1olem  24001  grpokerinj  26575  rngohom0  26603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-grpo 20858  df-gid 20859  df-ghom 21025
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