Theorem ablsub2inv 13961

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 13939	. . . 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 13694	. . . 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 13719	. 2 |- ( ph -> ( ( N ` X ) .- ( N ` Y ) ) = ( ( N ` X ) ( +g ` G ) Y ) )
13	1, 2	ablcom 13953	. . . . . 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 1274	. . . . 5 |- ( ph -> ( ( N ` X ) ( +g ` G ) Y ) = ( Y ( +g ` G ) ( N ` X ) ) )
15	1, 4	grpinvinv 13713	. . . . . . 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 6043	. . . . 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 13694	. . . . . 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 13724	. . . . 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 1274	. . . 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 13692	. . . . 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 5652	. . 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 13728	. . 3 |- ( ( G e. Grp /\ X e. B /\ Y e. B ) -> ( N ` ( X .- Y ) ) = ( Y .- X ) )
29	7, 8, 11, 28	syl3anc 1274	. 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 1398   e. wcel 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   Grpcgrp 13646   invgcminusg 13647   -gcsg 13648   Abelcabl 13935

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-minusg 13650  df-sbg 13651  df-cmn 13936  df-abl 13937

This theorem is referenced by: (None)