| Step |
Hyp |
Ref |
Expression |
| 1 |
|
itg2i1fseq.1 |
|- ( ph -> F e. MblFn ) |
| 2 |
|
itg2i1fseq.2 |
|- ( ph -> F : RR --> ( 0 [,) +oo ) ) |
| 3 |
|
itg2i1fseq.3 |
|- ( ph -> P : NN --> dom S.1 ) |
| 4 |
|
itg2i1fseq.4 |
|- ( ph -> A. n e. NN ( 0p oR <_ ( P ` n ) /\ ( P ` n ) oR <_ ( P ` ( n + 1 ) ) ) ) |
| 5 |
|
itg2i1fseq.5 |
|- ( ph -> A. x e. RR ( n e. NN |-> ( ( P ` n ) ` x ) ) ~~> ( F ` x ) ) |
| 6 |
|
itg2i1fseq.6 |
|- S = ( m e. NN |-> ( S.1 ` ( P ` m ) ) ) |
| 7 |
|
itg2i1fseq3.7 |
|- ( ph -> ( S.2 ` F ) e. RR ) |
| 8 |
|
icossicc |
|- ( 0 [,) +oo ) C_ ( 0 [,] +oo ) |
| 9 |
|
fss |
|- ( ( F : RR --> ( 0 [,) +oo ) /\ ( 0 [,) +oo ) C_ ( 0 [,] +oo ) ) -> F : RR --> ( 0 [,] +oo ) ) |
| 10 |
2 8 9
|
sylancl |
|- ( ph -> F : RR --> ( 0 [,] +oo ) ) |
| 11 |
10
|
adantr |
|- ( ( ph /\ k e. NN ) -> F : RR --> ( 0 [,] +oo ) ) |
| 12 |
3
|
ffvelcdmda |
|- ( ( ph /\ k e. NN ) -> ( P ` k ) e. dom S.1 ) |
| 13 |
1 2 3 4 5
|
itg2i1fseqle |
|- ( ( ph /\ k e. NN ) -> ( P ` k ) oR <_ F ) |
| 14 |
|
itg2ub |
|- ( ( F : RR --> ( 0 [,] +oo ) /\ ( P ` k ) e. dom S.1 /\ ( P ` k ) oR <_ F ) -> ( S.1 ` ( P ` k ) ) <_ ( S.2 ` F ) ) |
| 15 |
11 12 13 14
|
syl3anc |
|- ( ( ph /\ k e. NN ) -> ( S.1 ` ( P ` k ) ) <_ ( S.2 ` F ) ) |
| 16 |
1 2 3 4 5 6 7 15
|
itg2i1fseq2 |
|- ( ph -> S ~~> ( S.2 ` F ) ) |