Theorem eqgval 13429

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 13428	. . 3 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
6	5	breqd 4045	. 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 4794	. . . 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 1005	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. X )
10	9	elexd 2776	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. _V )
11		simpr2 1006	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. X )
12	11	elexd 2776	. . . 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 2766	. . . . . . . 8 |- x e. _V
15		vex 2766	. . . . . . . 8 |- y e. _V
16	14, 15	prss 3779	. . . . . . 7 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
17		eleq1 2259	. . . . . . . 8 |- ( x = A -> ( x e. X <-> A e. X ) )
18		eleq1 2259	. . . . . . . 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 5561	. . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
22		id 19	. . . . . . . 8 |- ( y = B -> y = B )
23	21, 22	oveqan12d 5944	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( N ` x ) .+ y ) = ( ( N ` A ) .+ B ) )
24	23	eleq1d 2265	. . . . . 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 982	. . . . 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 2196	. . . 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 4299	. . 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 917	. 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 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   C_ wss 3157   {cpr 3624   class class class wbr 4034   {copab 4094   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780   invgcminusg 13203   ~_QG cqg 13375

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

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 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-eqg 13378

This theorem is referenced by:  eqger  13430  eqglact  13431  eqgid  13432  eqgcpbl  13434  eqgabl  13536