Theorem eqgval 13809

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 13808	. . 3 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
6	5	breqd 4099	. 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 4857	. . . 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 1029	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. X )
10	9	elexd 2816	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. _V )
11		simpr2 1030	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. X )
12	11	elexd 2816	. . . 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 2805	. . . . . . . 8 |- x e. _V
15		vex 2805	. . . . . . . 8 |- y e. _V
16	14, 15	prss 3829	. . . . . . 7 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
17		eleq1 2294	. . . . . . . 8 |- ( x = A -> ( x e. X <-> A e. X ) )
18		eleq1 2294	. . . . . . . 8 |- ( y = B -> ( y e. X <-> B e. X ) )
19	17, 18	bi2anan9 610	. . . . . . 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 5639	. . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
22		id 19	. . . . . . . 8 |- ( y = B -> y = B )
23	21, 22	oveqan12d 6036	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( N ` x ) .+ y ) = ( ( N ` A ) .+ B ) )
24	23	eleq1d 2300	. . . . . 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 1006	. . . . 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 2231	. . . 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 4358	. . 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 923	. 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 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   {copab 4149   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   invgcminusg 13583   ~_QG cqg 13755

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-eqg 13758

This theorem is referenced by:  eqger  13810  eqglact  13811  eqgid  13812  eqgcpbl  13814  eqgabl  13916