Theorem eqgabl 13781

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 2207	. . 3 |- ( invg ` G ) = ( invg ` G )
3		eqid 2207	. . 3 |- ( +g ` G ) = ( +g ` G )
4		eqgabl.r	. . 3 |- .~ = ( G ~QG S )
5	1, 2, 3, 4	eqgval 13674	. 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 13740	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> A e. X )
10	1, 2	grpinvcl 13495	. . . . . . . 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 531	. . . . . . 7 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> B e. X )
13	1, 3	ablcom 13754	. . . . . . 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 1250	. . . . . 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 13493	. . . . . . 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 2243	. . . . 5 |- ( ( ( G e. Abel /\ S C_ X ) /\ ( A e. X /\ B e. X ) ) -> ( ( ( invg ` G ) ` A ) ( +g ` G ) B ) = ( B .- A ) )
19	18	eleq1d 2276	. . . 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 983	. . 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 983	. . 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 981   = wceq 1373   e. wcel 2178   C_ wss 3174   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   Grpcgrp 13447   invgcminusg 13448   -gcsg 13449   ~_QG cqg 13620   Abelcabl 13736

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-minusg 13451  df-sbg 13452  df-eqg 13623  df-cmn 13737  df-abl 13738

This theorem is referenced by:  qusecsub  13782  2idlcpblrng  14400  zndvds  14526