Definition df-eqg 14898

Description: Define the equivalence relation in a quotient ring or quotient group (where i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)

Assertion

Ref	Expression
df-eqg	|- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i ) } )

Distinct variable group:   i, r, x, y

Detailed syntax breakdown of Definition df-eqg

Step	Hyp	Ref	Expression
1		cqg 14895	. 2 class ~QG
2		vr	. . 3 set r
3		vi	. . 3 set i
4		cvv 2916	. . 3 class _V
5		vx	. . . . . . . 8 set x
6	5	cv 1648	. . . . . . 7 class x
7		vy	. . . . . . . 8 set y
8	7	cv 1648	. . . . . . 7 class y
9	6, 8	cpr 3775	. . . . . 6 class { x , y }
10	2	cv 1648	. . . . . . 7 class r
11		cbs 13424	. . . . . . 7 class Base
12	10, 11	cfv 5413	. . . . . 6 class ( Base ` r )
13	9, 12	wss 3280	. . . . 5 wff { x , y } C_ ( Base ` r )
14		cminusg 14641	. . . . . . . . 9 class inv g
15	10, 14	cfv 5413	. . . . . . . 8 class ( inv g ` r )
16	6, 15	cfv 5413	. . . . . . 7 class ( ( inv g ` r ) ` x )
17		cplusg 13484	. . . . . . . 8 class +g
18	10, 17	cfv 5413	. . . . . . 7 class ( +g ` r )
19	16, 8, 18	co 6040	. . . . . 6 class ( ( ( inv g ` r ) ` x ) ( +g ` r ) y )
20	3	cv 1648	. . . . . 6 class i
21	19, 20	wcel 1721	. . . . 5 wff ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i
22	13, 21	wa 359	. . . 4 wff ( { x , y } C_ ( Base ` r ) /\ ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i )
23	22, 5, 7	copab 4225	. . 3 class { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i ) }
24	2, 3, 4, 4, 23	cmpt2 6042	. 2 class ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
25	1, 24	wceq 1649	1 wff ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( inv g ` r ) ` x ) ( +g ` r ) y ) e. i ) } )

This definition is referenced by:  releqg  14942  eqgfval  14943