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Theorem grpsubpropd2 12903
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 997 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ph )
2 simp2 998 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  (
Base `  G )
)
3 grpsubpropd2.1 . . . . . . . 8  |-  ( ph  ->  B  =  ( Base `  G ) )
433ad2ant1 1018 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  B  =  (
Base `  G )
)
52, 4eleqtrrd 2257 . . . . . 6  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  a  e.  B
)
6 grpsubpropd2.3 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
763ad2ant1 1018 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  G  e.  Grp )
8 simp3 999 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  b  e.  (
Base `  G )
)
9 eqid 2177 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
10 eqid 2177 . . . . . . . . 9  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 12849 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  b  e.  ( Base `  G ) )  -> 
( ( invg `  G ) `  b
)  e.  ( Base `  G ) )
127, 8, 11syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( ( invg `  G ) `
 b )  e.  ( Base `  G
) )
1312, 4eleqtrrd 2257 . . . . . 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 ) )
1514oveqrspc2v 5897 . . . . . 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 1236 . . . . 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 ) )
18 eqid 2177 . . . . . . . . . . . . 13  |-  ( 0g
`  G )  =  ( 0g `  G
)
199, 18grpidcl 12832 . . . . . . . . . . . 12  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
206, 19syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  G
)  e.  ( Base `  G ) )
2120, 3eleqtrrd 2257 . . . . . . . . . 10  |-  ( ph  ->  ( 0g `  G
)  e.  B )
2217, 21basmexd 12512 . . . . . . . . 9  |-  ( ph  ->  H  e.  _V )
233, 17, 6, 22, 14grpinvpropdg 12873 . . . . . . . 8  |-  ( ph  ->  ( invg `  G )  =  ( invg `  H
) )
2423fveq1d 5514 . . . . . . 7  |-  ( ph  ->  ( ( invg `  G ) `  b
)  =  ( ( invg `  H
) `  b )
)
2524oveq2d 5886 . . . . . 6  |-  ( ph  ->  ( a ( +g  `  H ) ( ( invg `  G
) `  b )
)  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
26253ad2ant1 1018 . . . . 5  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  H ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2716, 26eqtrd 2210 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) )  =  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) )
2827mpoeq3dva 5934 . . 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
) ) ) )
293, 17eqtr3d 2212 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
30 mpoeq12 5930 . . . 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 ) ) ) )
3129, 29, 30syl2anc 411 . . 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
) ) ) )
3228, 31eqtrd 2210 . 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
) ) ) )
33 eqid 2177 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
34 eqid 2177 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
359, 33, 10, 34grpsubfvalg 12846 . . 3  |-  ( G  e.  Grp  ->  ( -g `  G )  =  ( a  e.  (
Base `  G ) ,  b  e.  ( Base `  G )  |->  ( a ( +g  `  G
) ( ( invg `  G ) `
 b ) ) ) )
366, 35syl 14 . 2  |-  ( ph  ->  ( -g `  G
)  =  ( a  e.  ( Base `  G
) ,  b  e.  ( Base `  G
)  |->  ( a ( +g  `  G ) ( ( invg `  G ) `  b
) ) ) )
37 eqid 2177 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
38 eqid 2177 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
39 eqid 2177 . . . 4  |-  ( invg `  H )  =  ( invg `  H )
40 eqid 2177 . . . 4  |-  ( -g `  H )  =  (
-g `  H )
4137, 38, 39, 40grpsubfvalg 12846 . . 3  |-  ( H  e.  _V  ->  ( -g `  H )  =  ( a  e.  (
Base `  H ) ,  b  e.  ( Base `  H )  |->  ( a ( +g  `  H
) ( ( invg `  H ) `
 b ) ) ) )
4222, 41syl 14 . 2  |-  ( ph  ->  ( -g `  H
)  =  ( a  e.  ( Base `  H
) ,  b  e.  ( Base `  H
)  |->  ( a ( +g  `  H ) ( ( invg `  H ) `  b
) ) ) )
4332, 36, 423eqtr4d 2220 1  |-  ( ph  ->  ( -g `  G
)  =  ( -g `  H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   _Vcvv 2737   ` cfv 5213  (class class class)co 5870    e. cmpo 5872   Basecbs 12452   +g cplusg 12526   0gc0g 12691   Grpcgrp 12805   invgcminusg 12806   -gcsg 12807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4116  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-cnex 7897  ax-resscn 7898  ax-1re 7900  ax-addrcl 7903
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-inn 8914  df-2 8972  df-ndx 12455  df-slot 12456  df-base 12458  df-plusg 12539  df-0g 12693  df-mgm 12705  df-sgrp 12738  df-mnd 12748  df-grp 12808  df-minusg 12809  df-sbg 12810
This theorem is referenced by: (None)
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