Theorem eqgval 13674

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 13673	. . 3 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
6	5	breqd 4070	. 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 4823	. . . 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 1006	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. X )
10	9	elexd 2790	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. _V )
11		simpr2 1007	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. X )
12	11	elexd 2790	. . . 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 2779	. . . . . . . 8 |- x e. _V
15		vex 2779	. . . . . . . 8 |- y e. _V
16	14, 15	prss 3800	. . . . . . 7 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
17		eleq1 2270	. . . . . . . 8 |- ( x = A -> ( x e. X <-> A e. X ) )
18		eleq1 2270	. . . . . . . 8 |- ( y = B -> ( y e. X <-> B e. X ) )
19	17, 18	bi2anan9 606	. . . . . . 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 5599	. . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
22		id 19	. . . . . . . 8 |- ( y = B -> y = B )
23	21, 22	oveqan12d 5986	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( N ` x ) .+ y ) = ( ( N ` A ) .+ B ) )
24	23	eleq1d 2276	. . . . . 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 983	. . . . 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 2207	. . . 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 4328	. . 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 918	. 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 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   {cpr 3644   class class class wbr 4059   {copab 4120   ` 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:  eqger  13675  eqglact  13676  eqgid  13677  eqgcpbl  13679  eqgabl  13781