Step |
Hyp |
Ref |
Expression |
1 |
|
geoserg.1 |
|- ( ph -> A e. CC ) |
2 |
|
geoserg.2 |
|- ( ph -> A =/= 1 ) |
3 |
|
geoserg.3 |
|- ( ph -> M e. NN0 ) |
4 |
|
geoserg.4 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
5 |
|
fzofi |
|- ( M ..^ N ) e. Fin |
6 |
5
|
a1i |
|- ( ph -> ( M ..^ N ) e. Fin ) |
7 |
|
ax-1cn |
|- 1 e. CC |
8 |
|
subcl |
|- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC ) |
9 |
7 1 8
|
sylancr |
|- ( ph -> ( 1 - A ) e. CC ) |
10 |
1
|
adantr |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> A e. CC ) |
11 |
|
elfzouz |
|- ( k e. ( M ..^ N ) -> k e. ( ZZ>= ` M ) ) |
12 |
|
eluznn0 |
|- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 ) |
13 |
3 11 12
|
syl2an |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> k e. NN0 ) |
14 |
10 13
|
expcld |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( A ^ k ) e. CC ) |
15 |
6 9 14
|
fsummulc1 |
|- ( ph -> ( sum_ k e. ( M ..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M ..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) ) |
16 |
7
|
a1i |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> 1 e. CC ) |
17 |
14 16 10
|
subdid |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) ) |
18 |
14
|
mulid1d |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) ) |
19 |
10 13
|
expp1d |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) ) |
20 |
19
|
eqcomd |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( ( A ^ k ) x. A ) = ( A ^ ( k + 1 ) ) ) |
21 |
18 20
|
oveq12d |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) ) |
22 |
17 21
|
eqtrd |
|- ( ( ph /\ k e. ( M ..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) ) |
23 |
22
|
sumeq2dv |
|- ( ph -> sum_ k e. ( M ..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M ..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) ) |
24 |
|
oveq2 |
|- ( j = k -> ( A ^ j ) = ( A ^ k ) ) |
25 |
|
oveq2 |
|- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) ) |
26 |
|
oveq2 |
|- ( j = M -> ( A ^ j ) = ( A ^ M ) ) |
27 |
|
oveq2 |
|- ( j = N -> ( A ^ j ) = ( A ^ N ) ) |
28 |
1
|
adantr |
|- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC ) |
29 |
|
elfzuz |
|- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) ) |
30 |
|
eluznn0 |
|- ( ( M e. NN0 /\ j e. ( ZZ>= ` M ) ) -> j e. NN0 ) |
31 |
3 29 30
|
syl2an |
|- ( ( ph /\ j e. ( M ... N ) ) -> j e. NN0 ) |
32 |
28 31
|
expcld |
|- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC ) |
33 |
24 25 26 27 4 32
|
telfsumo |
|- ( ph -> sum_ k e. ( M ..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) - ( A ^ N ) ) ) |
34 |
15 23 33
|
3eqtrrd |
|- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M ..^ N ) ( A ^ k ) x. ( 1 - A ) ) ) |
35 |
1 3
|
expcld |
|- ( ph -> ( A ^ M ) e. CC ) |
36 |
|
eluznn0 |
|- ( ( M e. NN0 /\ N e. ( ZZ>= ` M ) ) -> N e. NN0 ) |
37 |
3 4 36
|
syl2anc |
|- ( ph -> N e. NN0 ) |
38 |
1 37
|
expcld |
|- ( ph -> ( A ^ N ) e. CC ) |
39 |
35 38
|
subcld |
|- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC ) |
40 |
6 14
|
fsumcl |
|- ( ph -> sum_ k e. ( M ..^ N ) ( A ^ k ) e. CC ) |
41 |
2
|
necomd |
|- ( ph -> 1 =/= A ) |
42 |
|
subeq0 |
|- ( ( 1 e. CC /\ A e. CC ) -> ( ( 1 - A ) = 0 <-> 1 = A ) ) |
43 |
7 1 42
|
sylancr |
|- ( ph -> ( ( 1 - A ) = 0 <-> 1 = A ) ) |
44 |
43
|
necon3bid |
|- ( ph -> ( ( 1 - A ) =/= 0 <-> 1 =/= A ) ) |
45 |
41 44
|
mpbird |
|- ( ph -> ( 1 - A ) =/= 0 ) |
46 |
39 40 9 45
|
divmul3d |
|- ( ph -> ( ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( M ..^ N ) ( A ^ k ) <-> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M ..^ N ) ( A ^ k ) x. ( 1 - A ) ) ) ) |
47 |
34 46
|
mpbird |
|- ( ph -> ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( M ..^ N ) ( A ^ k ) ) |
48 |
47
|
eqcomd |
|- ( ph -> sum_ k e. ( M ..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) ) |