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Theorem ghomid 21048
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 2296 . . . . . . 7  |-  ran  G  =  ran  G
2 ghomid.1 . . . . . . 7  |-  U  =  (GId `  G )
31, 2grpoidcl 20900 . . . . . 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 21047 . . . 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 20902 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  U  e.  ran  G )  -> 
( U G U )  =  U )
93, 8mpdan 649 . . . . 5  |-  ( G  e.  GrpOp  ->  ( U G U )  =  U )
109fveq2d 5545 . . . 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 2328 . 2  |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  ( ( F `
 U ) H ( F `  U
) )  =  ( F `  U ) )
13 eqid 2296 . . . . . . 7  |-  ran  H  =  ran  H
141, 13elghom 21046 . . . . . 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 5679 . . . 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 20906 . . . . 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 1632    e. wcel 1696   A.wral 2556   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   GrpOpHom cghom 21040
This theorem is referenced by:  ghomf1olem  24016  grpokerinj  26678  rngohom0  26706
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-grpo 20874  df-gid 20875  df-ghom 21041
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