Step |
Hyp |
Ref |
Expression |
0 |
|
casa |
|- AssAlg |
1 |
|
vw |
|- w |
2 |
|
clmod |
|- LMod |
3 |
|
crg |
|- Ring |
4 |
2 3
|
cin |
|- ( LMod i^i Ring ) |
5 |
|
csca |
|- Scalar |
6 |
1
|
cv |
|- w |
7 |
6 5
|
cfv |
|- ( Scalar ` w ) |
8 |
|
vf |
|- f |
9 |
|
vr |
|- r |
10 |
|
cbs |
|- Base |
11 |
8
|
cv |
|- f |
12 |
11 10
|
cfv |
|- ( Base ` f ) |
13 |
|
vx |
|- x |
14 |
6 10
|
cfv |
|- ( Base ` w ) |
15 |
|
vy |
|- y |
16 |
|
cvsca |
|- .s |
17 |
6 16
|
cfv |
|- ( .s ` w ) |
18 |
|
vs |
|- s |
19 |
|
cmulr |
|- .r |
20 |
6 19
|
cfv |
|- ( .r ` w ) |
21 |
|
vt |
|- t |
22 |
9
|
cv |
|- r |
23 |
18
|
cv |
|- s |
24 |
13
|
cv |
|- x |
25 |
22 24 23
|
co |
|- ( r s x ) |
26 |
21
|
cv |
|- t |
27 |
15
|
cv |
|- y |
28 |
25 27 26
|
co |
|- ( ( r s x ) t y ) |
29 |
24 27 26
|
co |
|- ( x t y ) |
30 |
22 29 23
|
co |
|- ( r s ( x t y ) ) |
31 |
28 30
|
wceq |
|- ( ( r s x ) t y ) = ( r s ( x t y ) ) |
32 |
22 27 23
|
co |
|- ( r s y ) |
33 |
24 32 26
|
co |
|- ( x t ( r s y ) ) |
34 |
33 30
|
wceq |
|- ( x t ( r s y ) ) = ( r s ( x t y ) ) |
35 |
31 34
|
wa |
|- ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
36 |
35 21 20
|
wsbc |
|- [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
37 |
36 18 17
|
wsbc |
|- [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
38 |
37 15 14
|
wral |
|- A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
39 |
38 13 14
|
wral |
|- A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
40 |
39 9 12
|
wral |
|- A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
41 |
40 8 7
|
wsbc |
|- [. ( Scalar ` w ) / f ]. A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) |
42 |
41 1 4
|
crab |
|- { w e. ( LMod i^i Ring ) | [. ( Scalar ` w ) / f ]. A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) } |
43 |
0 42
|
wceq |
|- AssAlg = { w e. ( LMod i^i Ring ) | [. ( Scalar ` w ) / f ]. A. r e. ( Base ` f ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) [. ( .s ` w ) / s ]. [. ( .r ` w ) / t ]. ( ( ( r s x ) t y ) = ( r s ( x t y ) ) /\ ( x t ( r s y ) ) = ( r s ( x t y ) ) ) } |