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Theorem grpsubpropd2 14777
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 956 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 957 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 977 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2443 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 977 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 958 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2366 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2366 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
119, 10grpinvcl 14737 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( inv g `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( inv g `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2443 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( inv g `  G ) `
 b )  e.  B )
14 grpsubpropd2.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )
1514proplem 13802 . . . . . 6  |-  ( (
ph  /\  ( a  e.  B  /\  (
( inv g `  G ) `  b
)  e.  B ) )  ->  ( a
( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  G ) `
 b ) ) )
161, 5, 13, 15syl12anc 1181 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  G ) `
 b ) ) )
17 grpsubpropd2.2 . . . . . . . . 9  |-  ( ph  ->  B  =  ( Base `  H ) )
183, 17, 14grpinvpropd 14753 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
1918fveq1d 5634 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
2019oveq2d 5997 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
21203ad2ant1 977 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  H ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
2216, 21eqtrd 2398 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
2322mpt2eq3dva 6038 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( inv g `  G ) `
 b ) ) )  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
243, 17eqtr3d 2400 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
25 mpt2eq12 6034 . . . 4  |-  ( ( ( Base `  G
)  =  ( Base `  H )  /\  ( Base `  G )  =  ( Base `  H
) )  ->  (
a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) ) )
2624, 24, 25syl2anc 642 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) ) )
2723, 26eqtrd 2398 . 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
) ) ) )
28 eqid 2366 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
29 eqid 2366 . . 3  |-  ( -g `  G )  =  (
-g `  G )
309, 28, 10, 29grpsubfval 14734 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
31 eqid 2366 . . 3  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2366 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
33 eqid 2366 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
34 eqid 2366 . . 3  |-  ( -g `  H )  =  (
-g `  H )
3531, 32, 33, 34grpsubfval 14734 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
3627, 30, 353eqtr4g 2423 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   Basecbs 13356   +g cplusg 13416   Grpcgrp 14572   inv gcminusg 14573   -gcsg 14575
This theorem is referenced by:  ngppropd  18366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700  df-sbg 14701
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