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

Theorem grpsubpropd 14889
Description: Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
Hypotheses
Ref Expression
grpsubpropd.b  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
grpsubpropd.p  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
Assertion
Ref Expression
grpsubpropd  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )

Proof of Theorem grpsubpropd
Dummy variables  a 
b  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsubpropd.b . . 3  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
2 grpsubpropd.p . . . 4  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
3 eqidd 2437 . . . 4  |-  ( ph  ->  a  =  a )
4 eqidd 2437 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
52proplem3 13916 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
64, 1, 5grpinvpropd 14866 . . . . 5  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
76fveq1d 5730 . . . 4  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
82, 3, 7oveq123d 6102 . . 3  |-  ( ph  ->  ( a ( +g  `  G ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
91, 1, 8mpt2eq123dv 6136 . 2  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( inv g `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
10 eqid 2436 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2436 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2436 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
13 eqid 2436 . . 3  |-  ( -g `  G )  =  (
-g `  G )
1410, 11, 12, 13grpsubfval 14847 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
15 eqid 2436 . . 3  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2436 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2436 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
18 eqid 2436 . . 3  |-  ( -g `  H )  =  (
-g `  H )
1915, 16, 17, 18grpsubfval 14847 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
209, 14, 193eqtr4g 2493 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469   +g cplusg 13529   inv gcminusg 14686   -gcsg 14688
This theorem is referenced by:  tngngp2  18693  tngngp  18695  ply1divalg2  20061  zhmnrg  24351
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-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-1st 6349  df-2nd 6350  df-riota 6549  df-0g 13727  df-minusg 14813  df-sbg 14814
  Copyright terms: Public domain W3C validator