Theorem eqgfval 13012

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 2748	. . . 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 12512	. . . . . . 7 |- Base Fn _V
6		funfvex 5531	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
7	6	funfni 5315	. . . . . . 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 2264	. . . . 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 4142	. . 3 |- ( ( G e. V /\ S C_ X ) -> S e. _V )
13		xpexg 4739	. . . . 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 2740	. . . . . . . . 9 |- x e. _V
17		vex 2740	. . . . . . . . 9 |- y e. _V
18	16, 17	prss 3748	. . . . . . . 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 4276	. . . . . 6 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ { <. x , y >. | ( x e. X /\ y e. X ) }
21		df-xp 4631	. . . . . 6 |- ( X X. X ) = { <. x , y >. | ( x e. X /\ y e. X ) }
22	20, 21	sseqtrri 3190	. . . . 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 4142	. . 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 5518	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = ( Base ` G ) )
27	26, 4	eqtr4di 2228	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = X )
28	27	sseq2d 3185	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( { x , y } C_ ( Base ` g ) <-> { x , y } C_ X ) )
29	25	fveq2d 5518	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = ( +g ` G ) )
30		eqgval.p	. . . . . . . . 9 |- .+ = ( +g ` G )
31	29, 30	eqtr4di 2228	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = .+ )
32	25	fveq2d 5518	. . . . . . . . . 10 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = ( invg ` G ) )
33		eqgval.n	. . . . . . . . . 10 |- N = ( invg ` G )
34	32, 33	eqtr4di 2228	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = N )
35	34	fveq1d 5516	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( ( invg ` g ) ` x ) = ( N ` x ) )
36		eqidd 2178	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> y = y )
37	31, 35, 36	oveq123d 5893	. . . . . . 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 2248	. . . . . 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 4068	. . . 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 12963	. . . 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 6001	. . 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 1238	. 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 2222	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 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   {cpr 3593   {copab 4062   X. cxp 4623   Fn wfn 5210   ` cfv 5215  (class class class)co 5872   Basecbs 12454   +g cplusg 12528   invgcminusg 12810   ~_QG cqg 12960

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-inn 8916  df-ndx 12457  df-slot 12458  df-base 12460  df-eqg 12963

This theorem is referenced by:  eqgval  13013