Theorem ablsub2inv 13114

Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)

Hypotheses

Ref	Expression
ablsub2inv.b	|- B = ( Base ` G )
ablsub2inv.m	|- .- = ( -g ` G )
ablsub2inv.n	|- N = ( invg ` G )
ablsub2inv.g	|- ( ph -> G e. Abel )
ablsub2inv.x	|- ( ph -> X e. B )
ablsub2inv.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
ablsub2inv	|- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Proof of Theorem ablsub2inv

Step	Hyp	Ref	Expression
1		ablsub2inv.b	. . 3 |- B = ( Base ` G )
2		eqid 2177	. . 3 |- ( +g ` G ) = ( +g ` G )
3		ablsub2inv.m	. . 3 |- .- = ( -g ` G )
4		ablsub2inv.n	. . 3 |- N = ( invg ` G )
5		ablsub2inv.g	. . . 4 |- ( ph -> G e. Abel )
6		ablgrp 13093	. . . 4 |- ( G e. Abel -> G e. Grp )
7	5, 6	syl 14	. . 3 |- ( ph -> G e. Grp )
8		ablsub2inv.x	. . . 4 |- ( ph -> X e. B )
9	1, 4	grpinvcl 12921	. . . 4 |- ( ( G e. Grp /\ X e. B ) -> ( N ` X ) e. B )
10	7, 8, 9	syl2anc 411	. . 3 |- ( ph -> ( N ` X ) e. B )
11		ablsub2inv.y	. . 3 |- ( ph -> Y e. B )
12	1, 2, 3, 4, 7, 10, 11	grpsubinv 12943	. 2 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13	1, 2	ablcom 13106	. . . . . 6 |- ( ( G e. Abel /\ ( N ` X ) e. B /\ Y e. B ) -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
14	5, 10, 11, 13	syl3anc 1238	. . . . 5 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15	1, 4	grpinvinv 12937	. . . . . . 7 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y )
16	7, 11, 15	syl2anc 411	. . . . . 6 |- ( ph -> ( N ` ( N ` Y ) ) = Y )
17	16	oveq1d 5890	. . . . 5 |- ( ph -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
18	14, 17	eqtr4d 2213	. . . 4 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
19	1, 4	grpinvcl 12921	. . . . . 6 |- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B )
20	7, 11, 19	syl2anc 411	. . . . 5 |- ( ph -> ( N ` Y ) e. B )
21	1, 2, 4	grpinvadd 12948	. . . . 5 |- ( ( G e. Grp /\ X e. B /\ ( N ` Y ) e. B ) -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
22	7, 8, 20, 21	syl3anc 1238	. . . 4 |- ( ph -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
23	18, 22	eqtr4d 2213	. . 3 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
24	1, 2, 4, 3	grpsubval 12919	. . . . 5 |- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
25	8, 11, 24	syl2anc 411	. . . 4 |- ( ph -> ( X .- Y ) = ( X ( +g ` G ) ( N ` Y ) ) )
26	25	fveq2d 5520	. . 3 |- ( ph -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
27	23, 26	eqtr4d 2213	. 2 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X .- Y ) ) )
28	1, 3, 4	grpinvsub 12952	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29	7, 8, 11, 28	syl3anc 1238	. 2 |- ( ph -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
30	12, 27, 29	3eqtrd 2214	1 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   Grpcgrp 12877   invgcminusg 12878   -gcsg 12879   Abelcabl 13089

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-in1 614  ax-in2 615  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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-inn 8920  df-2 8978  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-0g 12707  df-mgm 12775  df-sgrp 12808  df-mnd 12818  df-grp 12880  df-minusg 12881  df-sbg 12882  df-cmn 13090  df-abl 13091

This theorem is referenced by: (None)