Theorem eqgfval 13352

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		eqgval.r	. 2 |- R = ( G ~QG S )
2		elex 2774	. . . 4 |- ( G e. V -> G e. _V )
3	2	adantr 276	. . 3 |- ( ( G e. V /\ S C_ X ) -> G e. _V )
4		eqgval.x	. . . . . 6 |- X = ( Base ` G )
5		basfn 12736	. . . . . . 7 |- Base Fn _V
6		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
7	6	funfni 5358	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
8	5, 2, 7	sylancr 414	. . . . . 6 |- ( G e. V -> ( Base ` G ) e. _V )
9	4, 8	eqeltrid 2283	. . . . 5 |- ( G e. V -> X e. _V )
10	9	adantr 276	. . . 4 |- ( ( G e. V /\ S C_ X ) -> X e. _V )
11		simpr 110	. . . 4 |- ( ( G e. V /\ S C_ X ) -> S C_ X )
12	10, 11	ssexd 4173	. . 3 |- ( ( G e. V /\ S C_ X ) -> S e. _V )
13		xpexg 4777	. . . . 5 |- ( ( X e. _V /\ X e. _V ) -> ( X X. X ) e. _V )
14	10, 10, 13	syl2anc 411	. . . 4 |- ( ( G e. V /\ S C_ X ) -> ( X X. X ) e. _V )
15		simpl 109	. . . . . . . 8 |- ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) -> { x , y } C_ X )
16		vex 2766	. . . . . . . . 9 |- x e. _V
17		vex 2766	. . . . . . . . 9 |- y e. _V
18	16, 17	prss 3778	. . . . . . . 8 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
19	15, 18	sylibr 134	. . . . . . 7 |- ( ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) -> ( x e. X /\ y e. X ) )
20	19	ssopab2i 4312	. . . . . 6 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ { <. x , y >. | ( x e. X /\ y e. X ) }
21		df-xp 4669	. . . . . 6 |- ( X X. X ) = { <. x , y >. | ( x e. X /\ y e. X ) }
22	20, 21	sseqtrri 3218	. . . . 5 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ ( X X. X )
23	22	a1i 9	. . . 4 |- ( ( G e. V /\ S C_ X ) -> { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ ( X X. X ) )
24	14, 23	ssexd 4173	. . 3 |- ( ( G e. V /\ S C_ X ) -> { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } e. _V )
25		simpl 109	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> g = G )
26	25	fveq2d 5562	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = ( Base ` G ) )
27	26, 4	eqtr4di 2247	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = X )
28	27	sseq2d 3213	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( { x , y } C_ ( Base ` g ) <-> { x , y } C_ X ) )
29	25	fveq2d 5562	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = ( +g ` G ) )
30		eqgval.p	. . . . . . . . 9 |- .+ = ( +g ` G )
31	29, 30	eqtr4di 2247	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = .+ )
32	25	fveq2d 5562	. . . . . . . . . 10 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = ( invg ` G ) )
33		eqgval.n	. . . . . . . . . 10 |- N = ( invg ` G )
34	32, 33	eqtr4di 2247	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = N )
35	34	fveq1d 5560	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( ( invg ` g ) ` x ) = ( N ` x ) )
36		eqidd 2197	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> y = y )
37	31, 35, 36	oveq123d 5943	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) = ( ( N ` x ) .+ y ) )
38		simpr 110	. . . . . . 7 |- ( ( g = G /\ s = S ) -> s = S )
39	37, 38	eleq12d 2267	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s <-> ( ( N ` x ) .+ y ) e. S ) )
40	28, 39	anbi12d 473	. . . . 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 ) ) )
41	40	opabbidv 4099	. . . 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 ) } )
42		df-eqg 13302	. . . 4 |- ~QG = ( g e. _V , s e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` g ) /\ ( ( ( invg ` g ) ` x ) ( +g ` g ) y ) e. s ) } )
43	41, 42	ovmpoga 6052	. . 3 |- ( ( G e. _V /\ S e. _V /\ { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } e. _V ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
44	3, 12, 24, 43	syl3anc 1249	. 2 |- ( ( G e. V /\ S C_ X ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
45	1, 44	eqtrid 2241	1 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   {cpr 3623   {copab 4093   X. cxp 4661   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   invgcminusg 13133   ~_QG cqg 13299

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-eqg 13302

This theorem is referenced by:  eqgval  13353