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Theorem grpsubpropd2 14853
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 957 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 958 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2489 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 959 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2412 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2412 . . . . . . . . 9  |-  ( inv g `  G )  =  ( inv g `  G )
119, 10grpinvcl 14813 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( inv g `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( inv g `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2489 . . . . . 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 13878 . . . . . 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 1182 . . . . 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 14829 . . . . . . . 8  |-  ( ph  ->  ( inv g `  G )  =  ( inv g `  H
) )
1918fveq1d 5697 . . . . . . 7  |-  ( ph  ->  ( ( inv g `  G ) `  b
)  =  ( ( inv g `  H
) `  b )
)
2019oveq2d 6064 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( inv g `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( inv g `  H ) `
 b ) ) )
21203ad2ant1 978 . . . . 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 2444 . . . 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 6105 . . 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 2446 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
25 mpt2eq12 6101 . . . 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 643 . . 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 2444 . 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 2412 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
29 eqid 2412 . . 3  |-  ( -g `  G )  =  (
-g `  G )
309, 28, 10, 29grpsubfval 14810 . 2  |-  ( -g `  G )  =  ( a  e.  ( Base `  G ) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( inv g `  G ) `  b
) ) )
31 eqid 2412 . . 3  |-  ( Base `  H )  =  (
Base `  H )
32 eqid 2412 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
33 eqid 2412 . . 3  |-  ( inv g `  H )  =  ( inv g `  H )
34 eqid 2412 . . 3  |-  ( -g `  H )  =  (
-g `  H )
3531, 32, 33, 34grpsubfval 14810 . 2  |-  ( -g `  H )  =  ( a  e.  ( Base `  H ) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( inv g `  H ) `  b
) ) )
3627, 30, 353eqtr4g 2469 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   Basecbs 13432   +g cplusg 13492   Grpcgrp 14648   inv gcminusg 14649   -gcsg 14651
This theorem is referenced by:  ngppropd  18639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777
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