Theorem eqgabl 14111

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 2238	. . 3 |- ( invg ` G ) = ( invg ` G )
3		eqid 2238	. . 3 |- ( +g ` G ) = ( +g ` G )
4		eqgabl.r	. . 3 |- .~ = ( G ~QG S )
5	1, 2, 3, 4	eqgval 14003	. 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 531	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Abel )
7		ablgrp 14069	. . . . . . . . 9 |- ( G e. Abel -> G e. Grp )
8	7	ad2antrr 492	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> G e. Grp )
9		simprl 535	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> A e. X )
10	1, 2	grpinvcl 13830	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( ( invg ` G ) ` A ) e. X )
11	8, 9, 10	syl2anc 415	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( invg ` G ) ` A ) e. X )
12		simprr 537	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> B e. X )
13	1, 3	ablcom 14083	. . . . . . 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 1278	. . . . . 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 13828	. . . . . . 7 |- ( ( B e. X /\ A e. X ) -> ( B .- A ) = ( B ( +g ` G ) ( ( invg ` G ) ` A ) ) )
17	12, 9, 16	syl2anc 415	. . . . . 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 2274	. . . . 5 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B .- A ) )
19	18	eleq1d 2307	. . . 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 456	. . 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 1011	. . 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 1011	. . 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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408   Grpcgrp 13782   invgcminusg 13783   -gcsg 13784   ~_QG cqg 13949   Abelcabl 14065

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-sbg 13787  df-eqg 13952  df-cmn 14066  df-abl 14067

This theorem is referenced by:  qusecsub  14112  2idlcpblrng  14832  zndvds  14956