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Theorem ghminv 14706
Description: A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Hypotheses
Ref Expression
ghminv.b  |-  B  =  ( Base `  S
)
ghminv.y  |-  M  =  ( inv g `  S )
ghminv.z  |-  N  =  ( inv g `  T )
Assertion
Ref Expression
ghminv  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  =  ( N `  ( F `  X )
) )

Proof of Theorem ghminv
StepHypRef Expression
1 ghmgrp1 14701 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 ghminv.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2296 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2296 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 ghminv.y . . . . . . 7  |-  M  =  ( inv g `  S )
62, 3, 4, 5grprinv 14545 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  S ) ( M `
 X ) )  =  ( 0g `  S ) )
71, 6sylan 457 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( X ( +g  `  S
) ( M `  X ) )  =  ( 0g `  S
) )
87fveq2d 5545 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( F `  ( 0g `  S ) ) )
92, 5grpinvcl 14543 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( M `  X
)  e.  B )
101, 9sylan 457 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( M `  X )  e.  B )
11 eqid 2296 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
122, 3, 11ghmlin 14704 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B  /\  ( M `  X )  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( ( F `  X ) ( +g  `  T ) ( F `
 ( M `  X ) ) ) )
1310, 12mpd3an3 1278 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( ( F `  X ) ( +g  `  T ) ( F `
 ( M `  X ) ) ) )
14 eqid 2296 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
154, 14ghmid 14705 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 451 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
178, 13, 163eqtr3d 2336 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( F `  X
) ( +g  `  T
) ( F `  ( M `  X ) ) )  =  ( 0g `  T ) )
18 ghmgrp2 14702 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
1918adantr 451 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  T  e.  Grp )
20 eqid 2296 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
212, 20ghmf 14703 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
22 ffvelrn 5679 . . . . 5  |-  ( ( F : B --> ( Base `  T )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  T
) )
2321, 22sylan 457 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  T
) )
2421adantr 451 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  F : B --> ( Base `  T
) )
25 ffvelrn 5679 . . . . 5  |-  ( ( F : B --> ( Base `  T )  /\  ( M `  X )  e.  B )  ->  ( F `  ( M `  X ) )  e.  ( Base `  T
) )
2624, 10, 25syl2anc 642 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  e.  ( Base `  T
) )
27 ghminv.z . . . . 5  |-  N  =  ( inv g `  T )
2820, 11, 14, 27grpinvid1 14546 . . . 4  |-  ( ( T  e.  Grp  /\  ( F `  X )  e.  ( Base `  T
)  /\  ( F `  ( M `  X
) )  e.  (
Base `  T )
)  ->  ( ( N `  ( F `  X ) )  =  ( F `  ( M `  X )
)  <->  ( ( F `
 X ) ( +g  `  T ) ( F `  ( M `  X )
) )  =  ( 0g `  T ) ) )
2919, 23, 26, 28syl3anc 1182 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( N `  ( F `  X )
)  =  ( F `
 ( M `  X ) )  <->  ( ( F `  X )
( +g  `  T ) ( F `  ( M `  X )
) )  =  ( 0g `  T ) ) )
3017, 29mpbird 223 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( N `  ( F `  X ) )  =  ( F `  ( M `  X )
) )
3130eqcomd 2301 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  =  ( N `  ( F `  X )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379    GrpHom cghm 14696
This theorem is referenced by:  ghmsub  14707  ghmmulg  14711  ghmrn  14712  ghmpreima  14720  ghmeql  14721  frgpup3lem  15102  mplind  16259  sum2dchr  20529
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-rmo 2564  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-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-ghm 14697
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