Theorem eqgval 13957

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 13956	. . 3 |- ( ( G e. V /\ S C_ X ) -> R = { <. x , y >. | ( { x , y } C_ X /\ ( ( N ` x ) .+ y ) e. S ) } )
6	5	breqd 4122	. 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 4884	. . . 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 1030	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. X )
10	9	elexd 2829	. . . 4 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> A e. _V )
11		simpr2 1031	. . . . 5 |- ( ( ( G e. V /\ S C_ X ) /\ ( A e. X /\ B e. X /\ ( ( N ` A ) .+ B ) e. S ) ) -> B e. X )
12	11	elexd 2829	. . . 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 2818	. . . . . . . 8 |- x e. _V
15		vex 2818	. . . . . . . 8 |- y e. _V
16	14, 15	prss 3852	. . . . . . 7 |- ( ( x e. X /\ y e. X ) <-> { x , y } C_ X )
17		eleq1 2297	. . . . . . . 8 |- ( x = A -> ( x e. X <-> A e. X ) )
18		eleq1 2297	. . . . . . . 8 |- ( y = B -> ( y e. X <-> B e. X ) )
19	17, 18	bi2anan9 610	. . . . . . 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 5672	. . . . . . . 8 |- ( x = A -> ( N ` x ) = ( N ` A ) )
22		id 19	. . . . . . . 8 |- ( y = B -> y = B )
23	21, 22	oveqan12d 6071	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( ( N ` x ) .+ y ) = ( ( N ` A ) .+ B ) )
24	23	eleq1d 2303	. . . . . 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 1007	. . . . 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 2234	. . . 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 4384	. . 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 924	. 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 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3213   {cpr 3692   class class class wbr 4111   {copab 4172   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   invgcminusg 13731   ~_QG cqg 13903

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1re 8223  ax-addrcl 8226

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-inn 9240  df-ndx 13232  df-slot 13233  df-base 13235  df-eqg 13906

This theorem is referenced by:  eqger  13958  eqglact  13959  eqgid  13960  eqgcpbl  13962  eqgabl  14064