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Theorem grpsubpropd 16339
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 2455 . . . 4  |-  ( ph  ->  a  =  a )
4 eqidd 2455 . . . . . 6  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
52oveqdr 6294 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  G
)  /\  y  e.  ( Base `  G )
) )  ->  (
x ( +g  `  G
) y )  =  ( x ( +g  `  H ) y ) )
64, 1, 5grpinvpropd 16312 . . . . 5  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
76fveq1d 5850 . . . 4  |-  ( ph  ->  ( ( invg `  G ) `  b
)  =  ( ( invg `  H
) `  b )
)
82, 3, 7oveq123d 6291 . . 3  |-  ( ph  ->  ( a ( +g  `  G ) ( ( invg `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
91, 1, 8mpt2eq123dv 6332 . 2  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
10 eqid 2454 . . 3  |-  ( Base `  G )  =  (
Base `  G )
11 eqid 2454 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
12 eqid 2454 . . 3  |-  ( invg `  G )  =  ( invg `  G )
13 eqid 2454 . . 3  |-  ( -g `  G )  =  (
-g `  G )
1410, 11, 12, 13grpsubfval 16291 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) ) )
15 eqid 2454 . . 3  |-  ( Base `  H )  =  (
Base `  H )
16 eqid 2454 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
17 eqid 2454 . . 3  |-  ( invg `  H )  =  ( invg `  H )
18 eqid 2454 . . 3  |-  ( -g `  H )  =  (
-g `  H )
1915, 16, 17, 18grpsubfval 16291 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) )
209, 14, 193eqtr4g 2520 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   Basecbs 14716   +g cplusg 14784   invgcminusg 16253   -gcsg 16254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-0g 14931  df-minusg 16257  df-sbg 16258
This theorem is referenced by:  rlmsub  18039  matsubg  19101  tngngp2  21332  tngngp  21334  tchsub  21830  ply1divalg2  22705  ttgsub  24384  zhmnrg  28182
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