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Theorem grpsubpropd 14566
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 2284 . . . 4  |-  ( ph  ->  a  =  a )
4 eqidd 2284 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
52proplem3 13593 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
64, 1, 5grpinvpropd 14543 . . . . 5  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
76fveq1d 5527 . . . 4  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
82, 3, 7oveq123d 5879 . . 3  |-  ( ph  ->  ( a ( +g  `  G ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
91, 1, 8mpt2eq123dv 5910 . 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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2283 . . 3  |-  ( inv g `  G )  =  ( inv g `  G )
13 eqid 2283 . . 3  |-  ( -g `  G )  =  (
-g `  G )
1410, 11, 12, 13grpsubfval 14524 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
15 eqid 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2283 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2283 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
18 eqid 2283 . . 3  |-  ( -g `  H )  =  (
-g `  H )
1915, 16, 17, 18grpsubfval 14524 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
209, 14, 193eqtr4g 2340 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208   inv gcminusg 14363   -gcsg 14365
This theorem is referenced by:  tngngp2  18168  tngngp  18170  ply1divalg2  19524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-minusg 14490  df-sbg 14491
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