Step |
Hyp |
Ref |
Expression |
1 |
|
ioombl1.b |
|- B = ( A (,) +oo ) |
2 |
|
ioombl1.a |
|- ( ph -> A e. RR ) |
3 |
|
ioombl1.e |
|- ( ph -> E C_ RR ) |
4 |
|
ioombl1.v |
|- ( ph -> ( vol* ` E ) e. RR ) |
5 |
|
ioombl1.c |
|- ( ph -> C e. RR+ ) |
6 |
|
ioombl1.s |
|- S = seq 1 ( + , ( ( abs o. - ) o. F ) ) |
7 |
|
ioombl1.t |
|- T = seq 1 ( + , ( ( abs o. - ) o. G ) ) |
8 |
|
ioombl1.u |
|- U = seq 1 ( + , ( ( abs o. - ) o. H ) ) |
9 |
|
ioombl1.f1 |
|- ( ph -> F : NN --> ( <_ i^i ( RR X. RR ) ) ) |
10 |
|
ioombl1.f2 |
|- ( ph -> E C_ U. ran ( (,) o. F ) ) |
11 |
|
ioombl1.f3 |
|- ( ph -> sup ( ran S , RR* , < ) <_ ( ( vol* ` E ) + C ) ) |
12 |
|
ioombl1.p |
|- P = ( 1st ` ( F ` n ) ) |
13 |
|
ioombl1.q |
|- Q = ( 2nd ` ( F ` n ) ) |
14 |
|
ioombl1.g |
|- G = ( n e. NN |-> <. if ( if ( P <_ A , A , P ) <_ Q , if ( P <_ A , A , P ) , Q ) , Q >. ) |
15 |
|
ioombl1.h |
|- H = ( n e. NN |-> <. P , if ( if ( P <_ A , A , P ) <_ Q , if ( P <_ A , A , P ) , Q ) >. ) |
16 |
|
eqid |
|- ( ( abs o. - ) o. F ) = ( ( abs o. - ) o. F ) |
17 |
16 6
|
ovolsf |
|- ( F : NN --> ( <_ i^i ( RR X. RR ) ) -> S : NN --> ( 0 [,) +oo ) ) |
18 |
9 17
|
syl |
|- ( ph -> S : NN --> ( 0 [,) +oo ) ) |
19 |
18
|
frnd |
|- ( ph -> ran S C_ ( 0 [,) +oo ) ) |
20 |
|
icossxr |
|- ( 0 [,) +oo ) C_ RR* |
21 |
19 20
|
sstrdi |
|- ( ph -> ran S C_ RR* ) |
22 |
|
supxrcl |
|- ( ran S C_ RR* -> sup ( ran S , RR* , < ) e. RR* ) |
23 |
21 22
|
syl |
|- ( ph -> sup ( ran S , RR* , < ) e. RR* ) |
24 |
5
|
rpred |
|- ( ph -> C e. RR ) |
25 |
4 24
|
readdcld |
|- ( ph -> ( ( vol* ` E ) + C ) e. RR ) |
26 |
|
mnfxr |
|- -oo e. RR* |
27 |
26
|
a1i |
|- ( ph -> -oo e. RR* ) |
28 |
18
|
ffnd |
|- ( ph -> S Fn NN ) |
29 |
|
1nn |
|- 1 e. NN |
30 |
|
fnfvelrn |
|- ( ( S Fn NN /\ 1 e. NN ) -> ( S ` 1 ) e. ran S ) |
31 |
28 29 30
|
sylancl |
|- ( ph -> ( S ` 1 ) e. ran S ) |
32 |
21 31
|
sseldd |
|- ( ph -> ( S ` 1 ) e. RR* ) |
33 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
34 |
|
ffvelrn |
|- ( ( S : NN --> ( 0 [,) +oo ) /\ 1 e. NN ) -> ( S ` 1 ) e. ( 0 [,) +oo ) ) |
35 |
18 29 34
|
sylancl |
|- ( ph -> ( S ` 1 ) e. ( 0 [,) +oo ) ) |
36 |
33 35
|
sselid |
|- ( ph -> ( S ` 1 ) e. RR ) |
37 |
36
|
mnfltd |
|- ( ph -> -oo < ( S ` 1 ) ) |
38 |
|
supxrub |
|- ( ( ran S C_ RR* /\ ( S ` 1 ) e. ran S ) -> ( S ` 1 ) <_ sup ( ran S , RR* , < ) ) |
39 |
21 31 38
|
syl2anc |
|- ( ph -> ( S ` 1 ) <_ sup ( ran S , RR* , < ) ) |
40 |
27 32 23 37 39
|
xrltletrd |
|- ( ph -> -oo < sup ( ran S , RR* , < ) ) |
41 |
|
xrre |
|- ( ( ( sup ( ran S , RR* , < ) e. RR* /\ ( ( vol* ` E ) + C ) e. RR ) /\ ( -oo < sup ( ran S , RR* , < ) /\ sup ( ran S , RR* , < ) <_ ( ( vol* ` E ) + C ) ) ) -> sup ( ran S , RR* , < ) e. RR ) |
42 |
23 25 40 11 41
|
syl22anc |
|- ( ph -> sup ( ran S , RR* , < ) e. RR ) |