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Theorem grpinvval2 15711
Description: A df-neg 9699-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 15668 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
4 eqid 2451 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
5 grpinvsub.n . . . 4  |-  N  =  ( invg `  G )
6 grpsubcl.m . . . 4  |-  .-  =  ( -g `  G )
71, 4, 5, 6grpsubval 15683 . . 3  |-  ( (  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
83, 7sylan 471 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
91, 5grpinvcl 15685 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
101, 4, 2grplid 15670 . . 3  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
119, 10syldan 470 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
128, 11eqtr2d 2493 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  =  (  .0.  .-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   0gc0g 14480   Grpcgrp 15512   invgcminusg 15513   -gcsg 15515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-sbg 15649
This theorem is referenced by:  istgp2  19778  nrmmetd  20283  nminv  20328  grpsubadd0sub  30904  matinvgcell  31007
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