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Theorem grpinvval2 16238
Description: A df-neg 9721-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 16195 . . 3  |-  ( G  e.  Grp  ->  .0.  e.  B )
4 eqid 2382 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
5 grpinvsub.n . . . 4  |-  N  =  ( invg `  G )
6 grpsubcl.m . . . 4  |-  .-  =  ( -g `  G )
71, 4, 5, 6grpsubval 16210 . . 3  |-  ( (  .0.  e.  B  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
83, 7sylan 469 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  .-  X
)  =  (  .0.  ( +g  `  G
) ( N `  X ) ) )
91, 5grpinvcl 16212 . . 3  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  e.  B )
101, 4, 2grplid 16197 . . 3  |-  ( ( G  e.  Grp  /\  ( N `  X )  e.  B )  -> 
(  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
119, 10syldan 468 . 2  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  ( +g  `  G ) ( N `
 X ) )  =  ( N `  X ) )
128, 11eqtr2d 2424 1  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( N `  X
)  =  (  .0.  .-  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171   -gcsg 16172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176
This theorem is referenced by:  grpsubadd0sub  16242  matinvgcell  19022  istgp2  20675  nrmmetd  21180  nminv  21225
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