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Theorem ghminv 16146
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  =  ( invg `  S )
ghminv.z  |-  N  =  ( invg `  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 16141 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 ghminv.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2467 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2467 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 ghminv.y . . . . . . 7  |-  M  =  ( invg `  S )
62, 3, 4, 5grprinv 15969 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  S ) ( M `
 X ) )  =  ( 0g `  S ) )
71, 6sylan 471 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( X ( +g  `  S
) ( M `  X ) )  =  ( 0g `  S
) )
87fveq2d 5876 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( F `  ( 0g `  S ) ) )
92, 5grpinvcl 15967 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( M `  X
)  e.  B )
101, 9sylan 471 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( M `  X )  e.  B )
11 eqid 2467 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
122, 3, 11ghmlin 16144 . . . . 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 1325 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( ( F `  X ) ( +g  `  T ) ( F `
 ( M `  X ) ) ) )
14 eqid 2467 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
154, 14ghmid 16145 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 465 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
178, 13, 163eqtr3d 2516 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( F `  X
) ( +g  `  T
) ( F `  ( M `  X ) ) )  =  ( 0g `  T ) )
18 ghmgrp2 16142 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
1918adantr 465 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  T  e.  Grp )
20 eqid 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
212, 20ghmf 16143 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
2221ffvelrnda 6032 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  T
) )
2321adantr 465 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  F : B --> ( Base `  T
) )
2423, 10ffvelrnd 6033 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  e.  ( Base `  T
) )
25 ghminv.z . . . . 5  |-  N  =  ( invg `  T )
2620, 11, 14, 25grpinvid1 15970 . . . 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 ) ) )
2719, 22, 24, 26syl3anc 1228 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( N `  ( F `  X )
)  =  ( F `
 ( M `  X ) )  <->  ( ( F `  X )
( +g  `  T ) ( F `  ( M `  X )
) )  =  ( 0g `  T ) ) )
2817, 27mpbird 232 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( N `  ( F `  X ) )  =  ( F `  ( M `  X )
) )
2928eqcomd 2475 1  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  =  ( N `  ( F `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926    GrpHom cghm 16136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-ghm 16137
This theorem is referenced by:  ghmsub  16147  ghmmulg  16151  ghmrn  16152  ghmpreima  16160  ghmeql  16161  frgpup3lem  16668  asclinvg  17860  mplind  18037  psgninv  18487  zrhpsgnodpm  18497  cpmatinvcl  19087  sum2dchr  23415
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