Theorem eqgex 13528

Description: The left coset equivalence relation exists. (Contributed by Jim Kingdon, 25-Apr-2025.)

Assertion

Ref	Expression
eqgex	|- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )

Proof of Theorem eqgex

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

Step	Hyp	Ref	Expression
1		elex 2782	. . . 4 |- ( G e. V -> G e. _V )
2	1	adantr 276	. . 3 |- ( ( G e. V /\ S e. W ) -> G e. _V )
3		elex 2782	. . . 4 |- ( S e. W -> S e. _V )
4	3	adantl 277	. . 3 |- ( ( G e. V /\ S e. W ) -> S e. _V )
5		vex 2774	. . . . . . 7 |- x e. _V
6		vex 2774	. . . . . . 7 |- y e. _V
7	5, 6	prss 3788	. . . . . 6 |- ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) <-> { x , y } C_ ( Base ` G ) )
8	7	anbi1i 458	. . . . 5 |- ( ( ( 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 ) )
9	8	opabbii 4110	. . . 4 |- { <. 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 ) }
10		basfn 12861	. . . . . . 7 |- Base Fn _V
11		funfvex 5592	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
12	11	funfni 5375	. . . . . . 7 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
13	10, 2, 12	sylancr 414	. . . . . 6 |- ( ( G e. V /\ S e. W ) -> ( Base ` G ) e. _V )
14		xpexg 4788	. . . . . 6 |- ( ( ( Base ` G ) e. _V /\ ( Base ` G ) e. _V ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
15	13, 13, 14	syl2anc 411	. . . . 5 |- ( ( G e. V /\ S e. W ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
16		opabssxp 4748	. . . . . 6 |- { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } C_ ( ( Base ` G ) X. ( Base ` G ) )
17	16	a1i 9	. . . . 5 |- ( ( 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 ) ) )
18	15, 17	ssexd 4183	. . . 4 |- ( ( 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 )
19	9, 18	eqeltrrid 2292	. . 3 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V )
20		fveq2 5575	. . . . . . 7 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
21	20	sseq2d 3222	. . . . . 6 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
22		fveq2 5575	. . . . . . . 8 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
23		fveq2 5575	. . . . . . . . 9 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
24	23	fveq1d 5577	. . . . . . . 8 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
25		eqidd 2205	. . . . . . . 8 |- ( r = G -> y = y )
26	22, 24, 25	oveq123d 5964	. . . . . . 7 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
27	26	eleq1d 2273	. . . . . 6 |- ( r = G -> ( ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) )
28	21, 27	anbi12d 473	. . . . 5 |- ( 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 ) ) )
29	28	opabbidv 4109	. . . 4 |- ( 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 ) } )
30		eleq2 2268	. . . . . 6 |- ( i = S -> ( ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) )
31	30	anbi2d 464	. . . . 5 |- ( 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 ) ) )
32	31	opabbidv 4109	. . . 4 |- ( 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 ) } )
33		df-eqg 13479	. . . 4 |- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
34	29, 32, 33	ovmpog 6079	. . 3 |- ( ( 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 ) } )
35	2, 4, 19, 34	syl3anc 1249	. 2 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
36	35, 19	eqeltrd 2281	1 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   {cpr 3633   {copab 4103   X. cxp 4672   Fn wfn 5265   ` cfv 5270  (class class class)co 5943   Basecbs 12803   +g cplusg 12880   invgcminusg 13304   ~_QG cqg 13476

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-inn 9036  df-ndx 12806  df-slot 12807  df-base 12809  df-eqg 13479

This theorem is referenced by:  quselbasg  13537  quseccl0g  13538  qusghm  13589  quscrng  14266  znval  14369  znle  14370  znbaslemnn  14372  znbas  14377  znzrhval  14380  znzrhfo  14381