MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ghminv Structured version   Unicode version

Theorem ghminv 15734
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 15729 . . . . . 6  |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
2 ghminv.b . . . . . . 7  |-  B  =  ( Base `  S
)
3 eqid 2433 . . . . . . 7  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2433 . . . . . . 7  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 ghminv.y . . . . . . 7  |-  M  =  ( invg `  S )
62, 3, 4, 5grprinv 15565 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  S ) ( M `
 X ) )  =  ( 0g `  S ) )
71, 6sylan 468 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( X ( +g  `  S
) ( M `  X ) )  =  ( 0g `  S
) )
87fveq2d 5683 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( F `  ( 0g `  S ) ) )
92, 5grpinvcl 15563 . . . . . 6  |-  ( ( S  e.  Grp  /\  X  e.  B )  ->  ( M `  X
)  e.  B )
101, 9sylan 468 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( M `  X )  e.  B )
11 eqid 2433 . . . . . 6  |-  ( +g  `  T )  =  ( +g  `  T )
122, 3, 11ghmlin 15732 . . . . 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 1308 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( X
( +g  `  S ) ( M `  X
) ) )  =  ( ( F `  X ) ( +g  `  T ) ( F `
 ( M `  X ) ) ) )
14 eqid 2433 . . . . . 6  |-  ( 0g
`  T )  =  ( 0g `  T
)
154, 14ghmid 15733 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  ( 0g `  S
) )  =  ( 0g `  T ) )
1615adantr 462 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( 0g `  S ) )  =  ( 0g `  T
) )
178, 13, 163eqtr3d 2473 . . 3  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  (
( F `  X
) ( +g  `  T
) ( F `  ( M `  X ) ) )  =  ( 0g `  T ) )
18 ghmgrp2 15730 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
1918adantr 462 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  T  e.  Grp )
20 eqid 2433 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
212, 20ghmf 15731 . . . . 5  |-  ( F  e.  ( S  GrpHom  T )  ->  F : B
--> ( Base `  T
) )
2221ffvelrnda 5831 . . . 4  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  ( F `  X )  e.  ( Base `  T
) )
2321adantr 462 . . . . 5  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  X  e.  B )  ->  F : B --> ( Base `  T
) )
2423, 10ffvelrnd 5832 . . . 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 15566 . . . 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 1211 . . 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 2438 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 1362    e. wcel 1755   -->wf 5402   ` cfv 5406  (class class class)co 6080   Basecbs 14157   +g cplusg 14221   0gc0g 14361   Grpcgrp 15393   invgcminusg 15394    GrpHom cghm 15724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-0g 14363  df-mnd 15398  df-grp 15525  df-minusg 15526  df-ghm 15725
This theorem is referenced by:  ghmsub  15735  ghmmulg  15739  ghmrn  15740  ghmpreima  15748  ghmeql  15749  frgpup3lem  16254  mplind  17516  psgninv  17854  zrhpsgnodpm  17864  sum2dchr  22498
  Copyright terms: Public domain W3C validator