Step |
Hyp |
Ref |
Expression |
1 |
|
effsumlt.1 |
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) ) |
2 |
|
effsumlt.2 |
|- ( ph -> A e. RR+ ) |
3 |
|
effsumlt.3 |
|- ( ph -> N e. NN0 ) |
4 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
5 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
6 |
1
|
eftval |
|- ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) ) |
7 |
6
|
adantl |
|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) ) |
8 |
2
|
rpred |
|- ( ph -> A e. RR ) |
9 |
|
reeftcl |
|- ( ( A e. RR /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR ) |
10 |
8 9
|
sylan |
|- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR ) |
11 |
7 10
|
eqeltrd |
|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR ) |
12 |
4 5 11
|
serfre |
|- ( ph -> seq 0 ( + , F ) : NN0 --> RR ) |
13 |
12 3
|
ffvelrnd |
|- ( ph -> ( seq 0 ( + , F ) ` N ) e. RR ) |
14 |
|
eqid |
|- ( ZZ>= ` ( N + 1 ) ) = ( ZZ>= ` ( N + 1 ) ) |
15 |
|
peano2nn0 |
|- ( N e. NN0 -> ( N + 1 ) e. NN0 ) |
16 |
3 15
|
syl |
|- ( ph -> ( N + 1 ) e. NN0 ) |
17 |
|
eqidd |
|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) = ( F ` k ) ) |
18 |
|
nn0z |
|- ( k e. NN0 -> k e. ZZ ) |
19 |
|
rpexpcl |
|- ( ( A e. RR+ /\ k e. ZZ ) -> ( A ^ k ) e. RR+ ) |
20 |
2 18 19
|
syl2an |
|- ( ( ph /\ k e. NN0 ) -> ( A ^ k ) e. RR+ ) |
21 |
|
faccl |
|- ( k e. NN0 -> ( ! ` k ) e. NN ) |
22 |
21
|
adantl |
|- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. NN ) |
23 |
22
|
nnrpd |
|- ( ( ph /\ k e. NN0 ) -> ( ! ` k ) e. RR+ ) |
24 |
20 23
|
rpdivcld |
|- ( ( ph /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. RR+ ) |
25 |
7 24
|
eqeltrd |
|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. RR+ ) |
26 |
8
|
recnd |
|- ( ph -> A e. CC ) |
27 |
1
|
efcllem |
|- ( A e. CC -> seq 0 ( + , F ) e. dom ~~> ) |
28 |
26 27
|
syl |
|- ( ph -> seq 0 ( + , F ) e. dom ~~> ) |
29 |
4 14 16 17 25 28
|
isumrpcl |
|- ( ph -> sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) e. RR+ ) |
30 |
13 29
|
ltaddrpd |
|- ( ph -> ( seq 0 ( + , F ) ` N ) < ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) ) |
31 |
1
|
efval2 |
|- ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) ) |
32 |
26 31
|
syl |
|- ( ph -> ( exp ` A ) = sum_ k e. NN0 ( F ` k ) ) |
33 |
11
|
recnd |
|- ( ( ph /\ k e. NN0 ) -> ( F ` k ) e. CC ) |
34 |
4 14 16 17 33 28
|
isumsplit |
|- ( ph -> sum_ k e. NN0 ( F ` k ) = ( sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) ) |
35 |
3
|
nn0cnd |
|- ( ph -> N e. CC ) |
36 |
|
ax-1cn |
|- 1 e. CC |
37 |
|
pncan |
|- ( ( N e. CC /\ 1 e. CC ) -> ( ( N + 1 ) - 1 ) = N ) |
38 |
35 36 37
|
sylancl |
|- ( ph -> ( ( N + 1 ) - 1 ) = N ) |
39 |
38
|
oveq2d |
|- ( ph -> ( 0 ... ( ( N + 1 ) - 1 ) ) = ( 0 ... N ) ) |
40 |
39
|
sumeq1d |
|- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = sum_ k e. ( 0 ... N ) ( F ` k ) ) |
41 |
|
eqidd |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( F ` k ) = ( F ` k ) ) |
42 |
3 4
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 0 ) ) |
43 |
|
elfznn0 |
|- ( k e. ( 0 ... N ) -> k e. NN0 ) |
44 |
43 33
|
sylan2 |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( F ` k ) e. CC ) |
45 |
41 42 44
|
fsumser |
|- ( ph -> sum_ k e. ( 0 ... N ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) ) |
46 |
40 45
|
eqtrd |
|- ( ph -> sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) = ( seq 0 ( + , F ) ` N ) ) |
47 |
46
|
oveq1d |
|- ( ph -> ( sum_ k e. ( 0 ... ( ( N + 1 ) - 1 ) ) ( F ` k ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) ) |
48 |
32 34 47
|
3eqtrd |
|- ( ph -> ( exp ` A ) = ( ( seq 0 ( + , F ) ` N ) + sum_ k e. ( ZZ>= ` ( N + 1 ) ) ( F ` k ) ) ) |
49 |
30 48
|
breqtrrd |
|- ( ph -> ( seq 0 ( + , F ) ` N ) < ( exp ` A ) ) |