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Theorem grpsubpropd 14582
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 2297 . . . 4  |-  ( ph  ->  a  =  a )
4 eqidd 2297 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
52proplem3 13609 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
64, 1, 5grpinvpropd 14559 . . . . 5  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
76fveq1d 5543 . . . 4  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
82, 3, 7oveq123d 5895 . . 3  |-  ( ph  ->  ( a ( +g  `  G ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
91, 1, 8mpt2eq123dv 5926 . 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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2296 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
13 eqid 2296 . . 3  |-  ( -g `  G )  =  (
-g `  G )
1410, 11, 12, 13grpsubfval 14540 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
15 eqid 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2296 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2296 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
18 eqid 2296 . . 3  |-  ( -g `  H )  =  (
-g `  H )
1915, 16, 17, 18grpsubfval 14540 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
209, 14, 193eqtr4g 2353 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   +g cplusg 13224   inv gcminusg 14379   -gcsg 14381
This theorem is referenced by:  tngngp2  18184  tngngp  18186  ply1divalg2  19540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-0g 13420  df-minusg 14506  df-sbg 14507
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