Step |
Hyp |
Ref |
Expression |
1 |
|
ssrab2 |
|- { x e. RR | ( x - B ) e. A } C_ RR |
2 |
1
|
a1i |
|- ( ( A e. dom vol /\ B e. RR ) -> { x e. RR | ( x - B ) e. A } C_ RR ) |
3 |
|
elpwi |
|- ( y e. ~P RR -> y C_ RR ) |
4 |
|
simpll |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> A e. dom vol ) |
5 |
|
ssrab2 |
|- { z e. RR | ( z - -u B ) e. y } C_ RR |
6 |
5
|
a1i |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> { z e. RR | ( z - -u B ) e. y } C_ RR ) |
7 |
|
simprl |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> y C_ RR ) |
8 |
|
simplr |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> B e. RR ) |
9 |
8
|
renegcld |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> -u B e. RR ) |
10 |
|
eqidd |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> { z e. RR | ( z - -u B ) e. y } = { z e. RR | ( z - -u B ) e. y } ) |
11 |
7 9 10
|
ovolshft |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` y ) = ( vol* ` { z e. RR | ( z - -u B ) e. y } ) ) |
12 |
|
simprr |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` y ) e. RR ) |
13 |
11 12
|
eqeltrrd |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` { z e. RR | ( z - -u B ) e. y } ) e. RR ) |
14 |
|
mblsplit |
|- ( ( A e. dom vol /\ { z e. RR | ( z - -u B ) e. y } C_ RR /\ ( vol* ` { z e. RR | ( z - -u B ) e. y } ) e. RR ) -> ( vol* ` { z e. RR | ( z - -u B ) e. y } ) = ( ( vol* ` ( { z e. RR | ( z - -u B ) e. y } i^i A ) ) + ( vol* ` ( { z e. RR | ( z - -u B ) e. y } \ A ) ) ) ) |
15 |
4 6 13 14
|
syl3anc |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` { z e. RR | ( z - -u B ) e. y } ) = ( ( vol* ` ( { z e. RR | ( z - -u B ) e. y } i^i A ) ) + ( vol* ` ( { z e. RR | ( z - -u B ) e. y } \ A ) ) ) ) |
16 |
|
inss1 |
|- ( y i^i { x e. RR | ( x - B ) e. A } ) C_ y |
17 |
16 7
|
sstrid |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( y i^i { x e. RR | ( x - B ) e. A } ) C_ RR ) |
18 |
|
mblss |
|- ( A e. dom vol -> A C_ RR ) |
19 |
4 18
|
syl |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> A C_ RR ) |
20 |
|
eqidd |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> { x e. RR | ( x - B ) e. A } = { x e. RR | ( x - B ) e. A } ) |
21 |
19 8 20
|
shft2rab |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> A = { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) |
22 |
21
|
ineq2d |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( { z e. RR | ( z - -u B ) e. y } i^i A ) = ( { z e. RR | ( z - -u B ) e. y } i^i { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) ) |
23 |
|
inrab |
|- ( { z e. RR | ( z - -u B ) e. y } i^i { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) = { z e. RR | ( ( z - -u B ) e. y /\ ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) } |
24 |
|
elin |
|- ( ( z - -u B ) e. ( y i^i { x e. RR | ( x - B ) e. A } ) <-> ( ( z - -u B ) e. y /\ ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) ) |
25 |
24
|
rabbii |
|- { z e. RR | ( z - -u B ) e. ( y i^i { x e. RR | ( x - B ) e. A } ) } = { z e. RR | ( ( z - -u B ) e. y /\ ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) } |
26 |
23 25
|
eqtr4i |
|- ( { z e. RR | ( z - -u B ) e. y } i^i { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) = { z e. RR | ( z - -u B ) e. ( y i^i { x e. RR | ( x - B ) e. A } ) } |
27 |
22 26
|
eqtrdi |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( { z e. RR | ( z - -u B ) e. y } i^i A ) = { z e. RR | ( z - -u B ) e. ( y i^i { x e. RR | ( x - B ) e. A } ) } ) |
28 |
17 9 27
|
ovolshft |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) = ( vol* ` ( { z e. RR | ( z - -u B ) e. y } i^i A ) ) ) |
29 |
7
|
ssdifssd |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( y \ { x e. RR | ( x - B ) e. A } ) C_ RR ) |
30 |
21
|
difeq2d |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( { z e. RR | ( z - -u B ) e. y } \ A ) = ( { z e. RR | ( z - -u B ) e. y } \ { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) ) |
31 |
|
difrab |
|- ( { z e. RR | ( z - -u B ) e. y } \ { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) = { z e. RR | ( ( z - -u B ) e. y /\ -. ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) } |
32 |
|
eldif |
|- ( ( z - -u B ) e. ( y \ { x e. RR | ( x - B ) e. A } ) <-> ( ( z - -u B ) e. y /\ -. ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) ) |
33 |
32
|
rabbii |
|- { z e. RR | ( z - -u B ) e. ( y \ { x e. RR | ( x - B ) e. A } ) } = { z e. RR | ( ( z - -u B ) e. y /\ -. ( z - -u B ) e. { x e. RR | ( x - B ) e. A } ) } |
34 |
31 33
|
eqtr4i |
|- ( { z e. RR | ( z - -u B ) e. y } \ { z e. RR | ( z - -u B ) e. { x e. RR | ( x - B ) e. A } } ) = { z e. RR | ( z - -u B ) e. ( y \ { x e. RR | ( x - B ) e. A } ) } |
35 |
30 34
|
eqtrdi |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( { z e. RR | ( z - -u B ) e. y } \ A ) = { z e. RR | ( z - -u B ) e. ( y \ { x e. RR | ( x - B ) e. A } ) } ) |
36 |
29 9 35
|
ovolshft |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) = ( vol* ` ( { z e. RR | ( z - -u B ) e. y } \ A ) ) ) |
37 |
28 36
|
oveq12d |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) = ( ( vol* ` ( { z e. RR | ( z - -u B ) e. y } i^i A ) ) + ( vol* ` ( { z e. RR | ( z - -u B ) e. y } \ A ) ) ) ) |
38 |
15 11 37
|
3eqtr4d |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ ( y C_ RR /\ ( vol* ` y ) e. RR ) ) -> ( vol* ` y ) = ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) ) |
39 |
38
|
expr |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ y C_ RR ) -> ( ( vol* ` y ) e. RR -> ( vol* ` y ) = ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) ) ) |
40 |
3 39
|
sylan2 |
|- ( ( ( A e. dom vol /\ B e. RR ) /\ y e. ~P RR ) -> ( ( vol* ` y ) e. RR -> ( vol* ` y ) = ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) ) ) |
41 |
40
|
ralrimiva |
|- ( ( A e. dom vol /\ B e. RR ) -> A. y e. ~P RR ( ( vol* ` y ) e. RR -> ( vol* ` y ) = ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) ) ) |
42 |
|
ismbl |
|- ( { x e. RR | ( x - B ) e. A } e. dom vol <-> ( { x e. RR | ( x - B ) e. A } C_ RR /\ A. y e. ~P RR ( ( vol* ` y ) e. RR -> ( vol* ` y ) = ( ( vol* ` ( y i^i { x e. RR | ( x - B ) e. A } ) ) + ( vol* ` ( y \ { x e. RR | ( x - B ) e. A } ) ) ) ) ) ) |
43 |
2 41 42
|
sylanbrc |
|- ( ( A e. dom vol /\ B e. RR ) -> { x e. RR | ( x - B ) e. A } e. dom vol ) |