Step |
Hyp |
Ref |
Expression |
0 |
|
cidl |
|- Idl |
1 |
|
vr |
|- r |
2 |
|
crngo |
|- RingOps |
3 |
|
vi |
|- i |
4 |
|
c1st |
|- 1st |
5 |
1
|
cv |
|- r |
6 |
5 4
|
cfv |
|- ( 1st ` r ) |
7 |
6
|
crn |
|- ran ( 1st ` r ) |
8 |
7
|
cpw |
|- ~P ran ( 1st ` r ) |
9 |
|
cgi |
|- GId |
10 |
6 9
|
cfv |
|- ( GId ` ( 1st ` r ) ) |
11 |
3
|
cv |
|- i |
12 |
10 11
|
wcel |
|- ( GId ` ( 1st ` r ) ) e. i |
13 |
|
vx |
|- x |
14 |
|
vy |
|- y |
15 |
13
|
cv |
|- x |
16 |
14
|
cv |
|- y |
17 |
15 16 6
|
co |
|- ( x ( 1st ` r ) y ) |
18 |
17 11
|
wcel |
|- ( x ( 1st ` r ) y ) e. i |
19 |
18 14 11
|
wral |
|- A. y e. i ( x ( 1st ` r ) y ) e. i |
20 |
|
vz |
|- z |
21 |
20
|
cv |
|- z |
22 |
|
c2nd |
|- 2nd |
23 |
5 22
|
cfv |
|- ( 2nd ` r ) |
24 |
21 15 23
|
co |
|- ( z ( 2nd ` r ) x ) |
25 |
24 11
|
wcel |
|- ( z ( 2nd ` r ) x ) e. i |
26 |
15 21 23
|
co |
|- ( x ( 2nd ` r ) z ) |
27 |
26 11
|
wcel |
|- ( x ( 2nd ` r ) z ) e. i |
28 |
25 27
|
wa |
|- ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) |
29 |
28 20 7
|
wral |
|- A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) |
30 |
19 29
|
wa |
|- ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) |
31 |
30 13 11
|
wral |
|- A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) |
32 |
12 31
|
wa |
|- ( ( GId ` ( 1st ` r ) ) e. i /\ A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) ) |
33 |
32 3 8
|
crab |
|- { i e. ~P ran ( 1st ` r ) | ( ( GId ` ( 1st ` r ) ) e. i /\ A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) ) } |
34 |
1 2 33
|
cmpt |
|- ( r e. RingOps |-> { i e. ~P ran ( 1st ` r ) | ( ( GId ` ( 1st ` r ) ) e. i /\ A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) ) } ) |
35 |
0 34
|
wceq |
|- Idl = ( r e. RingOps |-> { i e. ~P ran ( 1st ` r ) | ( ( GId ` ( 1st ` r ) ) e. i /\ A. x e. i ( A. y e. i ( x ( 1st ` r ) y ) e. i /\ A. z e. ran ( 1st ` r ) ( ( z ( 2nd ` r ) x ) e. i /\ ( x ( 2nd ` r ) z ) e. i ) ) ) } ) |