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Theorem ghminv 14690
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 14685 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 ghminv.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2283 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2283 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 ghminv.y . . . . . . 7  |-  M  =  ( inv g `  S )
62, 3, 4, 5grprinv 14529 . . . . . 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 5529 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( F `  ( 0g `  S ) ) )
92, 5grpinvcl 14527 . . . . . 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 2283 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
122, 3, 11ghmlin 14688 . . . . 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 2283 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
154, 14ghmid 14689 . . . . 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 2323 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( F `  X
) ( +g  `  T
) ( F `  ( M `  X ) ) )  =  ( 0g `  T ) )
18 ghmgrp2 14686 . . . . 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 2283 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
212, 20ghmf 14687 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
22 ffvelrn 5663 . . . . 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 5663 . . . . 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 14530 . . . 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 2288 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 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363    GrpHom cghm 14680
This theorem is referenced by:  ghmsub  14691  ghmmulg  14695  ghmrn  14696  ghmpreima  14704  ghmeql  14705  frgpup3lem  15086  mplind  16243  sum2dchr  20513
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-rmo 2551  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-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-ghm 14681
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