Theorem eqgabl 18240

Description: Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
eqgabl.x	|- X = ( Base ` G )
eqgabl.n	|- .- = ( -g ` G )
eqgabl.r	|- .~ = ( G ~QG S )

Assertion

Ref	Expression
eqgabl	|- ( ( G e. Abel /\ S C_ X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( B .- A ) e. S ) ) )

Proof of Theorem eqgabl

Step	Hyp	Ref	Expression
1		eqgabl.x	. . 3 |- X = ( Base ` G )
2		eqid 2622	. . 3 |- ( invg ` G ) = ( invg ` G )
3		eqid 2622	. . 3 |- ( +g ` G ) = ( +g ` G )
4		eqgabl.r	. . 3 |- .~ = ( G ~QG S )
5	1, 2, 3, 4	eqgval 17643	. 2 |- ( ( G e. Abel /\ S C_ X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S ) ) )
6		simpll 790	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Abel )
7		ablgrp 18198	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Grp )
9		simprl 794	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> A e. X )
10	1, 2	grpinvcl 17467	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
11	8, 9, 10	syl2anc 693	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( invg ` G ) ` A ) e. X )
12		simprr 796	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> B e. X )
13	1, 3	ablcom 18210	. . . . . . 7 |- ( ( G e. Abel /\ ( ( invg ` G ) ` A ) e. X /\ B e. X ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
14	6, 11, 12, 13	syl3anc 1326	. . . . . 6 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
15		eqgabl.n	. . . . . . . 8 |- .- = ( -g ` G )
16	1, 3, 2, 15	grpsubval 17465	. . . . . . 7 |- ( ( B e. X /\ A e. X ) -> ( B .- A ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
17	12, 9, 16	syl2anc 693	. . . . . 6 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( B .- A ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
18	14, 17	eqtr4d 2659	. . . . 5 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B .- A ) )
19	18	eleq1d 2686	. . . 4 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S <-> ( B .- A ) e. S ) )
20	19	pm5.32da 673	. . 3 |- ( ( G e. Abel /\ S C_ X ) -> ( ( ( A e. X /\ B e. X ) /\ ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S ) <-> ( ( A e. X /\ B e. X ) /\ ( B .- A ) e. S ) ) )
21		df-3an 1039	. . 3 |- ( ( A e. X /\ B e. X /\ ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S ) <-> ( ( A e. X /\ B e. X ) /\ ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S ) )
22		df-3an 1039	. . 3 |- ( ( A e. X /\ B e. X /\ ( B .- A ) e. S ) <-> ( ( A e. X /\ B e. X ) /\ ( B .- A ) e. S ) )
23	20, 21, 22	3bitr4g 303	. 2 |- ( ( G e. Abel /\ S C_ X ) -> ( ( A e. X /\ B e. X /\ ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) e. S ) <-> ( A e. X /\ B e. X /\ ( B .- A ) e. S ) ) )
24	5, 23	bitrd 268	1 |- ( ( G e. Abel /\ S C_ X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( B .- A ) e. S ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   ~_QG cqg 17590   Abelcabl 18194

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  df-eqg 17593  df-cmn 18195  df-abl 18196

This theorem is referenced by:  2idlcpbl  19234  zndvds  19898  tgptsmscls  21953