Theorem eqgex 13557

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 2783	. . . 4 |- ( G e. V -> G e. _V )
2	1	adantr 276	. . 3 |- ( ( G e. V /\ S e. W ) -> G e. _V )
3		elex 2783	. . . 4 |- ( S e. W -> S e. _V )
4	3	adantl 277	. . 3 |- ( ( G e. V /\ S e. W ) -> S e. _V )
5		vex 2775	. . . . . . 7 |- x e. _V
6		vex 2775	. . . . . . 7 |- y e. _V
7	5, 6	prss 3789	. . . . . 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 4111	. . . 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 12890	. . . . . . 7 |- Base Fn _V
11		funfvex 5593	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
12	11	funfni 5376	. . . . . . 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 4789	. . . . . 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 4749	. . . . . 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 4184	. . . 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 2293	. . 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 5576	. . . . . . 7 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
21	20	sseq2d 3223	. . . . . 6 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
22		fveq2 5576	. . . . . . . 8 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
23		fveq2 5576	. . . . . . . . 9 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
24	23	fveq1d 5578	. . . . . . . 8 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
25		eqidd 2206	. . . . . . . 8 |- ( r = G -> y = y )
26	22, 24, 25	oveq123d 5965	. . . . . . 7 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
27	26	eleq1d 2274	. . . . . 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 4110	. . . 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 2269	. . . . . 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 4110	. . . 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 13508	. . . 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 6080	. . 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 1250	. 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 2282	1 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   {cpr 3634   {copab 4104   X. cxp 4673   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   invgcminusg 13333   ~_QG cqg 13505

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-eqg 13508

This theorem is referenced by:  quselbasg  13566  quseccl0g  13567  qusghm  13618  quscrng  14295  znval  14398  znle  14399  znbaslemnn  14401  znbas  14406  znzrhval  14409  znzrhfo  14410