Step |
Hyp |
Ref |
Expression |
1 |
|
xrltso |
|- < Or RR* |
2 |
1
|
infex |
|- inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) e. _V |
3 |
|
df-ovol |
|- vol* = ( x e. ~P RR |-> inf ( { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( x C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) } , RR* , < ) ) |
4 |
2 3
|
fnmpti |
|- vol* Fn ~P RR |
5 |
|
elpwi |
|- ( x e. ~P RR -> x C_ RR ) |
6 |
|
ovolcl |
|- ( x C_ RR -> ( vol* ` x ) e. RR* ) |
7 |
|
ovolge0 |
|- ( x C_ RR -> 0 <_ ( vol* ` x ) ) |
8 |
|
pnfge |
|- ( ( vol* ` x ) e. RR* -> ( vol* ` x ) <_ +oo ) |
9 |
6 8
|
syl |
|- ( x C_ RR -> ( vol* ` x ) <_ +oo ) |
10 |
|
0xr |
|- 0 e. RR* |
11 |
|
pnfxr |
|- +oo e. RR* |
12 |
|
elicc1 |
|- ( ( 0 e. RR* /\ +oo e. RR* ) -> ( ( vol* ` x ) e. ( 0 [,] +oo ) <-> ( ( vol* ` x ) e. RR* /\ 0 <_ ( vol* ` x ) /\ ( vol* ` x ) <_ +oo ) ) ) |
13 |
10 11 12
|
mp2an |
|- ( ( vol* ` x ) e. ( 0 [,] +oo ) <-> ( ( vol* ` x ) e. RR* /\ 0 <_ ( vol* ` x ) /\ ( vol* ` x ) <_ +oo ) ) |
14 |
6 7 9 13
|
syl3anbrc |
|- ( x C_ RR -> ( vol* ` x ) e. ( 0 [,] +oo ) ) |
15 |
5 14
|
syl |
|- ( x e. ~P RR -> ( vol* ` x ) e. ( 0 [,] +oo ) ) |
16 |
15
|
rgen |
|- A. x e. ~P RR ( vol* ` x ) e. ( 0 [,] +oo ) |
17 |
|
ffnfv |
|- ( vol* : ~P RR --> ( 0 [,] +oo ) <-> ( vol* Fn ~P RR /\ A. x e. ~P RR ( vol* ` x ) e. ( 0 [,] +oo ) ) ) |
18 |
4 16 17
|
mpbir2an |
|- vol* : ~P RR --> ( 0 [,] +oo ) |