Theorem ablsub2inv 13897

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 2231	. . 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 13875	. . . 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 13630	. . . 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 13655	. 2 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13	1, 2	ablcom 13889	. . . . . 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 1273	. . . . 5 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15	1, 4	grpinvinv 13649	. . . . . . 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 6032	. . . . 5 |- ( ph -> ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) = ( Y ( +g ` G ) ( N ` X ) ) )
18	14, 17	eqtr4d 2267	. . . 4 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
19	1, 4	grpinvcl 13630	. . . . . 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 13660	. . . . 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 1273	. . . 4 |- ( ph -> ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) = ( ( N ` ( N ` Y ) ) ( +g ` G ) ( N ` X ) ) )
23	18, 22	eqtr4d 2267	. . 3 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
24	1, 2, 4, 3	grpsubval 13628	. . . . 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 5643	. . 3 |- ( ph -> ( N ` ( X .- Y ) ) = ( N ` ( X ( +g ` G ) ( N ` Y ) ) ) )
27	23, 26	eqtr4d 2267	. 2 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( N ` ( X .- Y ) ) )
28	1, 3, 4	grpinvsub 13664	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29	7, 8, 11, 28	syl3anc 1273	. 2 |- ( ph -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
30	12, 27, 29	3eqtrd 2268	1 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( Y .- X ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   Grpcgrp 13582   invgcminusg 13583   -gcsg 13584   Abelcabl 13871

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-sbg 13587  df-cmn 13872  df-abl 13873

This theorem is referenced by: (None)