| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fsump1i.1 |
|- Z = ( ZZ>= ` M ) |
| 2 |
|
fsump1i.2 |
|- N = ( K + 1 ) |
| 3 |
|
fsump1i.3 |
|- ( k = N -> A = B ) |
| 4 |
|
fsump1i.4 |
|- ( ( ph /\ k e. Z ) -> A e. CC ) |
| 5 |
|
fsump1i.5 |
|- ( ph -> ( K e. Z /\ sum_ k e. ( M ... K ) A = S ) ) |
| 6 |
|
fsump1i.6 |
|- ( ph -> ( S + B ) = T ) |
| 7 |
5
|
simpld |
|- ( ph -> K e. Z ) |
| 8 |
7 1
|
eleqtrdi |
|- ( ph -> K e. ( ZZ>= ` M ) ) |
| 9 |
|
peano2uz |
|- ( K e. ( ZZ>= ` M ) -> ( K + 1 ) e. ( ZZ>= ` M ) ) |
| 10 |
9 1
|
eleqtrrdi |
|- ( K e. ( ZZ>= ` M ) -> ( K + 1 ) e. Z ) |
| 11 |
8 10
|
syl |
|- ( ph -> ( K + 1 ) e. Z ) |
| 12 |
2 11
|
eqeltrid |
|- ( ph -> N e. Z ) |
| 13 |
2
|
oveq2i |
|- ( M ... N ) = ( M ... ( K + 1 ) ) |
| 14 |
13
|
sumeq1i |
|- sum_ k e. ( M ... N ) A = sum_ k e. ( M ... ( K + 1 ) ) A |
| 15 |
|
elfzuz |
|- ( k e. ( M ... ( K + 1 ) ) -> k e. ( ZZ>= ` M ) ) |
| 16 |
15 1
|
eleqtrrdi |
|- ( k e. ( M ... ( K + 1 ) ) -> k e. Z ) |
| 17 |
16 4
|
sylan2 |
|- ( ( ph /\ k e. ( M ... ( K + 1 ) ) ) -> A e. CC ) |
| 18 |
2
|
eqeq2i |
|- ( k = N <-> k = ( K + 1 ) ) |
| 19 |
18 3
|
sylbir |
|- ( k = ( K + 1 ) -> A = B ) |
| 20 |
8 17 19
|
fsump1 |
|- ( ph -> sum_ k e. ( M ... ( K + 1 ) ) A = ( sum_ k e. ( M ... K ) A + B ) ) |
| 21 |
14 20
|
eqtrid |
|- ( ph -> sum_ k e. ( M ... N ) A = ( sum_ k e. ( M ... K ) A + B ) ) |
| 22 |
5
|
simprd |
|- ( ph -> sum_ k e. ( M ... K ) A = S ) |
| 23 |
22
|
oveq1d |
|- ( ph -> ( sum_ k e. ( M ... K ) A + B ) = ( S + B ) ) |
| 24 |
21 23 6
|
3eqtrd |
|- ( ph -> sum_ k e. ( M ... N ) A = T ) |
| 25 |
12 24
|
jca |
|- ( ph -> ( N e. Z /\ sum_ k e. ( M ... N ) A = T ) ) |