Theorem eqgfval 17642

Description: Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-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
eqgfval	|- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )

Distinct variable groups:   x, y, G   x, N, y   x, S, y   x, .+ , y   x, X, y

Allowed substitution hints:   R( x, y)   V( x, y)

Proof of Theorem eqgfval

Dummy variables g s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( G e. V -> G e. _V )
2		eqgval.x	. . . 4 |- X = ( Base ` G )
3		fvex 6201	. . . 4 |- ( Base ` G ) e. _V
4	2, 3	eqeltri 2697	. . 3 |- X e. _V
5	4	ssex 4802	. 2 |- ( S C_ X -> S e. _V )
6		eqgval.r	. . 3 |- R = ( G ~QG S )
7		simpl 473	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> g = G )
8	7	fveq2d 6195	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = ( Base ` G ) )
9	8, 2	syl6eqr 2674	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = X )
10	9	sseq2d 3633	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( { x , y } C_ ( Base ` g ) <-> { x , y } C_ X ) )
11	7	fveq2d 6195	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = ( +g ` G ) )
12		eqgval.p	. . . . . . . . 9 |- .+ = ( +g ` G )
13	11, 12	syl6eqr 2674	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = .+ )
14	7	fveq2d 6195	. . . . . . . . . 10 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = ( invg ` G ) )
15		eqgval.n	. . . . . . . . . 10 |- N = ( invg ` G )
16	14, 15	syl6eqr 2674	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = N )
17	16	fveq1d 6193	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( ( invg ` g ) ` x ) = ( N ` x ) )
18		eqidd 2623	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> y = y )
19	13, 17, 18	oveq123d 6671	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) = ( ( N ` x ) .+ y ) )
20		simpr 477	. . . . . . 7 |- ( ( g = G /\ s = S ) -> s = S )
21	19, 20	eleq12d 2695	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s <-> ( ( N ` x ) .+ y ) e. S ) )
22	10, 21	anbi12d 747	. . . . 5 |- ( ( g = G /\ s = S ) -> ( ( { x , y } C_ ( Base ` g ) /\ ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s ) <-> ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) ) )
23	22	opabbidv 4716	. . . 4 |- ( ( g = G /\ s = S ) -> { <. x , y >. | ( { x , y } C_ ( Base ` g ) /\ ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s ) } = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
24		df-eqg 17593	. . . 4 |- ~QG = ( g e. _V , s e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` g ) /\ ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s ) } )
25	4, 4	xpex 6962	. . . . 5 |- ( X X. X ) e. _V
26		simpl 473	. . . . . . . 8 |- ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) -> { x , y } C_ X )
27		vex 3203	. . . . . . . . 9 |- x e. _V
28		vex 3203	. . . . . . . . 9 |- y e. _V
29	27, 28	prss 4351	. . . . . . . 8 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
30	26, 29	sylibr 224	. . . . . . 7 |- ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) -> ( x e. X /\ y e. X ) )
31	30	ssopab2i 5003	. . . . . 6 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ { <. x , y >. | ( x e. X /\ y e. X ) }
32		df-xp 5120	. . . . . 6 |- ( X X. X ) = { <. x , y >. | ( x e. X /\ y e. X ) }
33	31, 32	sseqtr4i 3638	. . . . 5 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ ( X X. X )
34	25, 33	ssexi 4803	. . . 4 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } e. _V
35	23, 24, 34	ovmpt2a 6791	. . 3 |- ( ( G e. _V /\ S e. _V ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
36	6, 35	syl5eq 2668	. 2 |- ( ( G e. _V /\ S e. _V ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
37	1, 5, 36	syl2an 494	1 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {cpr 4179   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   invgcminusg 17423   ~_QG cqg 17590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-eqg 17593

This theorem is referenced by:  eqgval  17643