Theorem releqgg 13973

Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypothesis

Ref	Expression
releqg.r	|- R = ( G ~QG S )

Assertion

Ref	Expression
releqgg	|- ( ( G e. V /\ S e. W ) -> Rel R )

Proof of Theorem releqgg

Dummy variables i r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4886	. 2 |- Rel { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) }
2		releqg.r	. . . 4 |- R = ( G ~QG S )
3		elex 2827	. . . . . 6 |- ( G e. V -> G e. _V )
4	3	adantr 276	. . . . 5 |- ( ( G e. V /\ S e. W ) -> G e. _V )
5		elex 2827	. . . . . 6 |- ( S e. W -> S e. _V )
6	5	adantl 277	. . . . 5 |- ( ( G e. V /\ S e. W ) -> S e. _V )
7		vex 2818	. . . . . . . . 9 |- x e. _V
8		vex 2818	. . . . . . . . 9 |- y e. _V
9	7, 8	prss 3855	. . . . . . . 8 |- ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) <-> { x , y } C_ ( Base ` G ) )
10	9	anbi1i 458	. . . . . . 7 |- ( ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) )
11	10	opabbii 4182	. . . . . 6 |- { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) }
12		basfn 13355	. . . . . . . . 9 |- Base Fn _V
13		funfvex 5692	. . . . . . . . . 10 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5463	. . . . . . . . 9 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
15	12, 4, 14	sylancr 414	. . . . . . . 8 |- ( ( G e. V /\ S e. W ) -> ( Base ` G ) e. _V )
16		xpexg 4869	. . . . . . . 8 |- ( ( ( Base ` G ) e. _V /\ ( Base ` G ) e. _V ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
17	15, 15, 16	syl2anc 411	. . . . . . 7 |- ( ( G e. V /\ S e. W ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
18		opabssxp 4829	. . . . . . . 8 |- { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } C_ ( ( Base ` G ) X. ( Base ` G ) )
19	18	a1i 9	. . . . . . 7 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } C_ ( ( Base ` G ) X. ( Base ` G ) ) )
20	17, 19	ssexd 4255	. . . . . 6 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V )
21	11, 20	eqeltrrid 2322	. . . . 5 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V )
22		fveq2 5675	. . . . . . . . 9 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
23	22	sseq2d 3272	. . . . . . . 8 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
24		fveq2 5675	. . . . . . . . . 10 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
25		fveq2 5675	. . . . . . . . . . 11 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
26	25	fveq1d 5677	. . . . . . . . . 10 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
27		eqidd 2235	. . . . . . . . . 10 |- ( r = G -> y = y )
28	24, 26, 27	oveq123d 6079	. . . . . . . . 9 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
29	28	eleq1d 2303	. . . . . . . 8 |- ( r = G -> ( ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) )
30	23, 29	anbi12d 473	. . . . . . 7 |- ( r = G -> ( ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) ) )
31	30	opabbidv 4181	. . . . . 6 |- ( r = G -> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) } )
32		eleq2 2298	. . . . . . . 8 |- ( i = S -> ( ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) )
33	32	anbi2d 464	. . . . . . 7 |- ( i = S -> ( ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) ) )
34	33	opabbidv 4181	. . . . . 6 |- ( i = S -> { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
35		df-eqg 13925	. . . . . 6 |- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
36	31, 34, 35	ovmpog 6196	. . . . 5 |- ( ( G e. _V /\ S e. _V /\ { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
37	4, 6, 21, 36	syl3anc 1274	. . . 4 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
38	2, 37	eqtrid 2279	. . 3 |- ( ( G e. V /\ S e. W ) -> R = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
39	38	releqd 4839	. 2 |- ( ( G e. V /\ S e. W ) -> ( Rel R <-> Rel { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } ) )
40	1, 39	mpbiri 168	1 |- ( ( G e. V /\ S e. W ) -> Rel R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   C_ wss 3214   {cpr 3695   {copab 4175   X. cxp 4752   Rel wrel 4759   Fn wfn 5352   ` cfv 5357  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   invgcminusg 13756   ~_QG cqg 13922

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

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 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-inn 9255  df-ndx 13299  df-slot 13300  df-base 13302  df-eqg 13925

This theorem is referenced by:  eqger  13977  eqgid  13979