Theorem grpinvsub 17497

Description: Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)

Hypotheses

Ref	Expression
grpsubcl.b	|- B = ( Base ` G )
grpsubcl.m	|- .- = ( -g ` G )
grpinvsub.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvsub	|- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )

Proof of Theorem grpinvsub

Step	Hyp	Ref	Expression
1		grpsubcl.b	. . . . . 6 |- B = ( Base ` G )
2		grpinvsub.n	. . . . . 6 |- N = ( invg ` G )
3	1, 2	grpinvcl 17467	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
4	3	3adant2 1080	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` Y ) e. B )
5		eqid 2622	. . . . 5 |- ( +g ` G ) = ( +g ` G )
6	1, 5, 2	grpinvadd 17493	. . . 4 |- ( ( G e. Grp /\ X e. B /\ ( N ` Y ) e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
7	4, 6	syld3an3 1371	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
8	1, 2	grpinvinv 17482	. . . . 5 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
9	8	3adant2 1080	. . . 4 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
10	9	oveq1d 6665	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
11	7, 10	eqtrd 2656	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
12		grpsubcl.m	. . . . 5 |- .- = ( -g ` G )
13	1, 5, 2, 12	grpsubval 17465	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
14	13	3adant1 1079	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
15	14	fveq2d 6195	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
16	1, 5, 2, 12	grpsubval 17465	. . . 4 |- ( ( Y e. B /\ X e. B ) -> ( Y .- X ) = ( Y ( +g ` G ) ( N ` X ) ) )
17	16	ancoms 469	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( Y .- X ) = ( Y ( +g ` G ) ( N ` X ) ) )
18	17	3adant1 1079	. 2 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( Y .- X ) = ( Y ( +g ` G ) ( N ` X ) ) )
19	11, 15, 18	3eqtr4d 2666	1 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427

This theorem is referenced by:  grpsubsub  17504  ablsub2inv  18216  lspsnsub  19007  ghmcnp  21918  nrmmetd  22379  nmsub  22427  mapdpglem14  36974