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Theorem grpsubpropd2 15937
Description: Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubpropd2.1  |-  ( ph  ->  B  =  ( Base `  G ) )
grpsubpropd2.2  |-  ( ph  ->  B  =  ( Base `  H ) )
grpsubpropd2.3  |-  ( ph  ->  G  e.  Grp )
grpsubpropd2.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
Assertion
Ref Expression
grpsubpropd2  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Distinct variable groups:    x, y, B    x, G, y    x, H, y    ph, x, y

Proof of Theorem grpsubpropd2
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 991 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 992 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 1012 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2553 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 1012 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 993 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2462 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2462 . . . . . . . . 9  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 15891 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( invg `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( invg `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2553 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( invg `  G ) `
 b )  e.  B )
14 grpsubpropd2.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
1514proplem 14936 . . . . . 6  |-  ( (
ph  /\  ( a  e.  B  /\  (
( invg `  G ) `  b
)  e.  B ) )  ->  ( a
( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  G ) `
 b ) ) )
161, 5, 13, 15syl12anc 1221 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  G ) `
 b ) ) )
17 grpsubpropd2.2 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  H ) )
183, 17, 14grpinvpropd 15909 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
1918fveq1d 5861 . . . . . . 7  |-  ( ph  ->  ( ( invg `  G ) `  b
)  =  ( ( invg `  H
) `  b )
)
2019oveq2d 6293 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( invg `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
21203ad2ant1 1012 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  H ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2216, 21eqtrd 2503 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2322mpt2eq3dva 6338 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
243, 17eqtr3d 2505 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
25 mpt2eq12 6334 . . . 4  |-  ( ( ( Base `  G
)  =  ( Base `  H )  /\  ( Base `  G )  =  ( Base `  H
) )  ->  (
a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) ) )
2624, 24, 25syl2anc 661 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
2723, 26eqtrd 2503 . 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
) ) ) )
28 eqid 2462 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
29 eqid 2462 . . 3  |-  ( -g `  G )  =  (
-g `  G )
309, 28, 10, 29grpsubfval 15888 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) ) )
31 eqid 2462 . . 3  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2462 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
33 eqid 2462 . . 3  |-  ( invg `  H )  =  ( invg `  H )
34 eqid 2462 . . 3  |-  ( -g `  H )  =  (
-g `  H )
3531, 32, 33, 34grpsubfval 15888 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) )
3627, 30, 353eqtr4g 2528 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14481   +g cplusg 14546   Grpcgrp 15718   invgcminusg 15719   -gcsg 15721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-0g 14688  df-mnd 15723  df-grp 15853  df-minusg 15854  df-sbg 15855
This theorem is referenced by:  ngppropd  20881
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