| 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 |
|
ffvelcdm |
|- ( ( 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 ) |