Step |
Hyp |
Ref |
Expression |
1 |
|
isumrecl.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
isumrecl.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
isumrecl.3 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A ) |
4 |
|
isumrecl.4 |
|- ( ( ph /\ k e. Z ) -> A e. RR ) |
5 |
|
isumrecl.5 |
|- ( ph -> seq M ( + , F ) e. dom ~~> ) |
6 |
|
isumge0.6 |
|- ( ( ph /\ k e. Z ) -> 0 <_ A ) |
7 |
4
|
recnd |
|- ( ( ph /\ k e. Z ) -> A e. CC ) |
8 |
1 2 3 7 5
|
isumclim2 |
|- ( ph -> seq M ( + , F ) ~~> sum_ k e. Z A ) |
9 |
|
fveq2 |
|- ( j = k -> ( F ` j ) = ( F ` k ) ) |
10 |
9
|
cbvsumv |
|- sum_ j e. Z ( F ` j ) = sum_ k e. Z ( F ` k ) |
11 |
3
|
sumeq2dv |
|- ( ph -> sum_ k e. Z ( F ` k ) = sum_ k e. Z A ) |
12 |
10 11
|
eqtrid |
|- ( ph -> sum_ j e. Z ( F ` j ) = sum_ k e. Z A ) |
13 |
8 12
|
breqtrrd |
|- ( ph -> seq M ( + , F ) ~~> sum_ j e. Z ( F ` j ) ) |
14 |
3 4
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) |
15 |
6 3
|
breqtrrd |
|- ( ( ph /\ k e. Z ) -> 0 <_ ( F ` k ) ) |
16 |
1 2 13 14 15
|
iserge0 |
|- ( ph -> 0 <_ sum_ j e. Z ( F ` j ) ) |
17 |
16 12
|
breqtrd |
|- ( ph -> 0 <_ sum_ k e. Z A ) |