Theorem ablsub2inv 13381

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 2193	. . 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 13359	. . . 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 13120	. . . 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 13145	. 2 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13	1, 2	ablcom 13373	. . . . . 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 1249	. . . . 5 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15	1, 4	grpinvinv 13139	. . . . . . 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 5933	. . . . 5 |- ( ph -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
18	14, 17	eqtr4d 2229	. . . 4 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
19	1, 4	grpinvcl 13120	. . . . . 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 13150	. . . . 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 1249	. . . 4 |- ( ph -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
23	18, 22	eqtr4d 2229	. . 3 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
24	1, 2, 4, 3	grpsubval 13118	. . . . 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 5558	. . 3 |- ( ph -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
27	23, 26	eqtr4d 2229	. 2 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X .- Y ) ) )
28	1, 3, 4	grpinvsub 13154	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29	7, 8, 11, 28	syl3anc 1249	. 2 |- ( ph -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
30	12, 27, 29	3eqtrd 2230	1 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   Grpcgrp 13072   invgcminusg 13073   -gcsg 13074   Abelcabl 13355

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-sbg 13077  df-cmn 13356  df-abl 13357

This theorem is referenced by: (None)