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Theorem grpinvval2 15602
Description: A df-neg 9594-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
grpsubcl.b  |-  B  =  ( Base `  G
)
grpsubcl.m  |-  .-  =  ( -g `  G )
grpinvsub.n  |-  N  =  ( invg `  G )
grpinvval2.z  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
grpinvval2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  =  (  .0.  .-  X ) )

Proof of Theorem grpinvval2
StepHypRef Expression
1 grpsubcl.b . . . 4  |-  B  =  ( Base `  G
)
2 grpinvval2.z . . . 4  |-  .0.  =  ( 0g `  G )
31, 2grpidcl 15559 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
4 eqid 2441 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
5 grpinvsub.n . . . 4  |-  N  =  ( invg `  G )
6 grpsubcl.m . . . 4  |-  .-  =  ( -g `  G )
71, 4, 5, 6grpsubval 15574 . . 3  |-  ( (  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
83, 7sylan 468 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
91, 5grpinvcl 15576 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
101, 4, 2grplid 15561 . . 3  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
119, 10syldan 467 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
128, 11eqtr2d 2474 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  =  (  .0.  .-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407   -gcsg 15409
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540
This theorem is referenced by:  istgp2  19621  nrmmetd  20126  nminv  20171
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