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Theorem ghminv 15013
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 15008 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 ghminv.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2436 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2436 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 ghminv.y . . . . . . 7  |-  M  =  ( inv g `  S )
62, 3, 4, 5grprinv 14852 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  S ) ( M `
 X ) )  =  ( 0g `  S ) )
71, 6sylan 458 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( X ( +g  `  S
) ( M `  X ) )  =  ( 0g `  S
) )
87fveq2d 5732 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( F `  ( 0g `  S ) ) )
92, 5grpinvcl 14850 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( M `  X
)  e.  B )
101, 9sylan 458 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( M `  X )  e.  B )
11 eqid 2436 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
122, 3, 11ghmlin 15011 . . . . 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 1280 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( ( F `  X ) ( +g  `  T ) ( F `
 ( M `  X ) ) ) )
14 eqid 2436 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
154, 14ghmid 15012 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 452 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
178, 13, 163eqtr3d 2476 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( F `  X
) ( +g  `  T
) ( F `  ( M `  X ) ) )  =  ( 0g `  T ) )
18 ghmgrp2 15009 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
1918adantr 452 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  T  e.  Grp )
20 eqid 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
212, 20ghmf 15010 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
2221ffvelrnda 5870 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  T
) )
2321adantr 452 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  F : B --> ( Base `  T
) )
2423, 10ffvelrnd 5871 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  e.  ( Base `  T
) )
25 ghminv.z . . . . 5  |-  N  =  ( inv g `  T )
2620, 11, 14, 25grpinvid1 14853 . . . 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 1184 . . 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 224 . 2  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( N `  ( F `  X ) )  =  ( F `  ( M `  X )
) )
2928eqcomd 2441 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   inv gcminusg 14686    GrpHom cghm 15003
This theorem is referenced by:  ghmsub  15014  ghmmulg  15018  ghmrn  15019  ghmpreima  15027  ghmeql  15028  frgpup3lem  15409  mplind  16562  sum2dchr  21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-ghm 15004
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