Theorem eqgfval 13673

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 2788	. . . 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 13005	. . . . . . 7 |- Base Fn _V
6		funfvex 5616	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
7	6	funfni 5395	. . . . . . 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 2294	. . . . 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 4200	. . 3 |- ( ( G e. V /\ S C_ X ) -> S e. _V )
13		xpexg 4807	. . . . 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 2779	. . . . . . . . 9 |- x e. _V
17		vex 2779	. . . . . . . . 9 |- y e. _V
18	16, 17	prss 3800	. . . . . . . 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 4342	. . . . . 6 |- { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } C_ { <. x , y >. | ( x e. X /\ y e. X ) }
21		df-xp 4699	. . . . . 6 |- ( X X. X ) = { <. x , y >. | ( x e. X /\ y e. X ) }
22	20, 21	sseqtrri 3236	. . . . 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 4200	. . 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 5603	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = ( Base ` G ) )
27	26, 4	eqtr4di 2258	. . . . . . 7 |- ( ( g = G /\ s = S ) -> ( Base ` g ) = X )
28	27	sseq2d 3231	. . . . . 6 |- ( ( g = G /\ s = S ) -> ( { x , y } C_ ( Base ` g ) <-> { x , y } C_ X ) )
29	25	fveq2d 5603	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = ( +g ` G ) )
30		eqgval.p	. . . . . . . . 9 |- .+ = ( +g ` G )
31	29, 30	eqtr4di 2258	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( +g ` g ) = .+ )
32	25	fveq2d 5603	. . . . . . . . . 10 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = ( invg ` G ) )
33		eqgval.n	. . . . . . . . . 10 |- N = ( invg ` G )
34	32, 33	eqtr4di 2258	. . . . . . . . 9 |- ( ( g = G /\ s = S ) -> ( invg ` g ) = N )
35	34	fveq1d 5601	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> ( ( invg ` g ) ` x ) = ( N ` x ) )
36		eqidd 2208	. . . . . . . 8 |- ( ( g = G /\ s = S ) -> y = y )
37	31, 35, 36	oveq123d 5988	. . . . . . 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 2278	. . . . . 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 4126	. . . 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 13623	. . . 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 6098	. . 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 1250	. 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 2252	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 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {cpr 3644   {copab 4120   X. cxp 4691   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   invgcminusg 13448   ~_QG cqg 13620

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-eqg 13623

This theorem is referenced by:  eqgval  13674