Theorem eqgval 13293

Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
eqgval.x	|- X = ( Base ` G )
eqgval.n	|- N = ( invg ` G )
eqgval.p	|- .+ = ( +g ` G )
eqgval.r	|- R = ( G ~QG S )

Assertion

Ref	Expression
eqgval	|- ( ( G e. V /\ S C_ X ) -> ( A R B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )

Proof of Theorem eqgval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqgval.x	. . . 4 |- X = ( Base ` G )
2		eqgval.n	. . . 4 |- N = ( invg ` G )
3		eqgval.p	. . . 4 |- .+ = ( +g ` G )
4		eqgval.r	. . . 4 |- R = ( G ~QG S )
5	1, 2, 3, 4	eqgfval 13292	. . 3 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
6	5	breqd 4040	. 2 |- ( ( G e. V /\ S C_ X ) -> ( A R B <-> A { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } B ) )
7		brabv 4789	. . . 4 |- ( A { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } B -> ( A e. _V /\ B e. _V ) )
8	7	adantl 277	. . 3 |- ( ( ( G e. V /\ S C_ X ) /\ A { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } B ) -> ( A e. _V /\ B e. _V ) )
9		simpr1 1005	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. X )
10	9	elexd 2773	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. _V )
11		simpr2 1006	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. X )
12	11	elexd 2773	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. _V )
13	10, 12	jca 306	. . 3 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> ( A e. _V /\ B e. _V ) )
14		vex 2763	. . . . . . . 8 |- x e. _V
15		vex 2763	. . . . . . . 8 |- y e. _V
16	14, 15	prss 3774	. . . . . . 7 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
17		eleq1 2256	. . . . . . . 8 |- ( x = A -> ( x e. X <-> A e. X ) )
18		eleq1 2256	. . . . . . . 8 |- ( y = B -> ( y e. X <-> B e. X ) )
19	17, 18	bi2anan9 606	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( x e. X /\ y e. X ) <-> ( A e. X /\ B e. X ) ) )
20	16, 19	bitr3id 194	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( { x , y } C_ X <-> ( A e. X /\ B e. X ) ) )
21		fveq2 5554	. . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
22		id 19	. . . . . . . 8 |- ( y = B -> y = B )
23	21, 22	oveqan12d 5937	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( N ` x ) .+ y ) = ( ( N ` A ) .+ B ) )
24	23	eleq1d 2262	. . . . . 6 |- ( ( x = A /\ y = B ) -> ( ( ( N ` x ) .+ y ) e. S <-> ( ( N ` A ) .+ B ) e. S ) )
25	20, 24	anbi12d 473	. . . . 5 |- ( ( x = A /\ y = B ) -> ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) <-> ( ( A e. X /\ B e. X ) /\ ( ( N ` A ) .+ B ) e. S ) ) )
26		df-3an 982	. . . . 5 |- ( ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) <-> ( ( A e. X /\ B e. X ) /\ ( ( N ` A ) .+ B ) e. S ) )
27	25, 26	bitr4di 198	. . . 4 |- ( ( x = A /\ y = B ) -> ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )
28		eqid 2193	. . . 4 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) }
29	27, 28	brabga 4294	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )
30	8, 13, 29	pm5.21nd 917	. 2 |- ( ( G e. V /\ S C_ X ) -> ( A { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )
31	6, 30	bitrd 188	1 |- ( ( G e. V /\ S C_ X ) -> ( A R B <-> ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {cpr 3619   class class class wbr 4029   {copab 4089   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   invgcminusg 13073   ~_QG cqg 13239

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-eqg 13242

This theorem is referenced by:  eqger  13294  eqglact  13295  eqgid  13296  eqgcpbl  13298  eqgabl  13400