Step |
Hyp |
Ref |
Expression |
0 |
|
cvol |
|- vol |
1 |
|
covol |
|- vol* |
2 |
|
vx |
|- x |
3 |
|
vy |
|- y |
4 |
1
|
ccnv |
|- `' vol* |
5 |
|
cr |
|- RR |
6 |
4 5
|
cima |
|- ( `' vol* " RR ) |
7 |
3
|
cv |
|- y |
8 |
7 1
|
cfv |
|- ( vol* ` y ) |
9 |
2
|
cv |
|- x |
10 |
7 9
|
cin |
|- ( y i^i x ) |
11 |
10 1
|
cfv |
|- ( vol* ` ( y i^i x ) ) |
12 |
|
caddc |
|- + |
13 |
7 9
|
cdif |
|- ( y \ x ) |
14 |
13 1
|
cfv |
|- ( vol* ` ( y \ x ) ) |
15 |
11 14 12
|
co |
|- ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) |
16 |
8 15
|
wceq |
|- ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) |
17 |
16 3 6
|
wral |
|- A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) |
18 |
17 2
|
cab |
|- { x | A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) } |
19 |
1 18
|
cres |
|- ( vol* |` { x | A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) } ) |
20 |
0 19
|
wceq |
|- vol = ( vol* |` { x | A. y e. ( `' vol* " RR ) ( vol* ` y ) = ( ( vol* ` ( y i^i x ) ) + ( vol* ` ( y \ x ) ) ) } ) |