Theorem eqgabl 14064

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 2234	. . 3 |- ( invg ` G ) = ( invg ` G )
3		eqid 2234	. . 3 |- ( +g ` G ) = ( +g ` G )
4		eqgabl.r	. . 3 |- .~ = ( G ~QG S )
5	1, 2, 3, 4	eqgval 13957	. 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 527	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Abel )
7		ablgrp 14023	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8	7	ad2antrr 488	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Grp )
9		simprl 531	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> A e. X )
10	1, 2	grpinvcl 13778	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
11	8, 9, 10	syl2anc 411	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( invg ` G ) ` A ) e. X )
12		simprr 533	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> B e. X )
13	1, 3	ablcom 14037	. . . . . . 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 1274	. . . . . 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 13776	. . . . . . 7 |- ( ( B e. X /\ A e. X ) -> ( B .- A ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
17	12, 9, 16	syl2anc 411	. . . . . 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 2270	. . . . 5 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B .- A ) )
19	18	eleq1d 2303	. . . 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 452	. . 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 1007	. . 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 1007	. . 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 223	. 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 188	1 |- ( ( G e. Abel /\ S C_ X ) -> ( A .~ B <-> ( A e. X /\ B e. X /\ ( B .- A ) e. S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3213   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   Grpcgrp 13730   invgcminusg 13731   -gcsg 13732   ~_QG cqg 13903   Abelcabl 14019

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-inn 9240  df-2 9298  df-ndx 13232  df-slot 13233  df-base 13235  df-plusg 13320  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-eqg 13906  df-cmn 14020  df-abl 14021

This theorem is referenced by:  qusecsub  14065  2idlcpblrng  14688  zndvds  14814