Theorem eqgex 13871

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 2815	. . . 4 |- ( G e. V -> G e. _V )
2	1	adantr 276	. . 3 |- ( ( G e. V /\ S e. W ) -> G e. _V )
3		elex 2815	. . . 4 |- ( S e. W -> S e. _V )
4	3	adantl 277	. . 3 |- ( ( G e. V /\ S e. W ) -> S e. _V )
5		vex 2806	. . . . . . 7 |- x e. _V
6		vex 2806	. . . . . . 7 |- y e. _V
7	5, 6	prss 3834	. . . . . 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 4161	. . . 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 13204	. . . . . . 7 |- Base Fn _V
11		funfvex 5665	. . . . . . . 8 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
12	11	funfni 5439	. . . . . . 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 4846	. . . . . 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 4806	. . . . . 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 4234	. . . 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 2319	. . 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 5648	. . . . . . 7 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
21	20	sseq2d 3258	. . . . . 6 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
22		fveq2 5648	. . . . . . . 8 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
23		fveq2 5648	. . . . . . . . 9 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
24	23	fveq1d 5650	. . . . . . . 8 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
25		eqidd 2232	. . . . . . . 8 |- ( r = G -> y = y )
26	22, 24, 25	oveq123d 6049	. . . . . . 7 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
27	26	eleq1d 2300	. . . . . 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 4160	. . . 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 2295	. . . . . 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 4160	. . . 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 13822	. . . 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 6166	. . 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 1274	. 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 2308	1 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   {cpr 3674   {copab 4154   X. cxp 4729   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   invgcminusg 13647   ~_QG cqg 13819

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-eqg 13822

This theorem is referenced by:  quselbasg  13880  quseccl0g  13881  qusghm  13932  quscrng  14612  znval  14715  znle  14716  znbaslemnn  14718  znbas  14723  znzrhval  14726  znzrhfo  14727