Step |
Hyp |
Ref |
Expression |
0 |
|
crngo |
|- RingOps |
1 |
|
vg |
|- g |
2 |
|
vh |
|- h |
3 |
1
|
cv |
|- g |
4 |
|
cablo |
|- AbelOp |
5 |
3 4
|
wcel |
|- g e. AbelOp |
6 |
2
|
cv |
|- h |
7 |
3
|
crn |
|- ran g |
8 |
7 7
|
cxp |
|- ( ran g X. ran g ) |
9 |
8 7 6
|
wf |
|- h : ( ran g X. ran g ) --> ran g |
10 |
5 9
|
wa |
|- ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) |
11 |
|
vx |
|- x |
12 |
|
vy |
|- y |
13 |
|
vz |
|- z |
14 |
11
|
cv |
|- x |
15 |
12
|
cv |
|- y |
16 |
14 15 6
|
co |
|- ( x h y ) |
17 |
13
|
cv |
|- z |
18 |
16 17 6
|
co |
|- ( ( x h y ) h z ) |
19 |
15 17 6
|
co |
|- ( y h z ) |
20 |
14 19 6
|
co |
|- ( x h ( y h z ) ) |
21 |
18 20
|
wceq |
|- ( ( x h y ) h z ) = ( x h ( y h z ) ) |
22 |
15 17 3
|
co |
|- ( y g z ) |
23 |
14 22 6
|
co |
|- ( x h ( y g z ) ) |
24 |
14 17 6
|
co |
|- ( x h z ) |
25 |
16 24 3
|
co |
|- ( ( x h y ) g ( x h z ) ) |
26 |
23 25
|
wceq |
|- ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) |
27 |
14 15 3
|
co |
|- ( x g y ) |
28 |
27 17 6
|
co |
|- ( ( x g y ) h z ) |
29 |
24 19 3
|
co |
|- ( ( x h z ) g ( y h z ) ) |
30 |
28 29
|
wceq |
|- ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) |
31 |
21 26 30
|
w3a |
|- ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) |
32 |
31 13 7
|
wral |
|- A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) |
33 |
32 12 7
|
wral |
|- A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) |
34 |
33 11 7
|
wral |
|- A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) |
35 |
16 15
|
wceq |
|- ( x h y ) = y |
36 |
15 14 6
|
co |
|- ( y h x ) |
37 |
36 15
|
wceq |
|- ( y h x ) = y |
38 |
35 37
|
wa |
|- ( ( x h y ) = y /\ ( y h x ) = y ) |
39 |
38 12 7
|
wral |
|- A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) |
40 |
39 11 7
|
wrex |
|- E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) |
41 |
34 40
|
wa |
|- ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) |
42 |
10 41
|
wa |
|- ( ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) /\ ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) ) |
43 |
42 1 2
|
copab |
|- { <. g , h >. | ( ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) /\ ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) ) } |
44 |
0 43
|
wceq |
|- RingOps = { <. g , h >. | ( ( g e. AbelOp /\ h : ( ran g X. ran g ) --> ran g ) /\ ( A. x e. ran g A. y e. ran g A. z e. ran g ( ( ( x h y ) h z ) = ( x h ( y h z ) ) /\ ( x h ( y g z ) ) = ( ( x h y ) g ( x h z ) ) /\ ( ( x g y ) h z ) = ( ( x h z ) g ( y h z ) ) ) /\ E. x e. ran g A. y e. ran g ( ( x h y ) = y /\ ( y h x ) = y ) ) ) } |