Step |
Hyp |
Ref |
Expression |
1 |
|
fprodefsum.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
fprodefsum.2 |
|- ( ph -> N e. Z ) |
3 |
|
fprodefsum.3 |
|- ( ( ph /\ k e. Z ) -> A e. CC ) |
4 |
2 1
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
5 |
|
oveq2 |
|- ( a = M -> ( M ... a ) = ( M ... M ) ) |
6 |
5
|
prodeq1d |
|- ( a = M -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
7 |
5
|
sumeq1d |
|- ( a = M -> sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) |
8 |
7
|
fveq2d |
|- ( a = M -> ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) ) |
9 |
6 8
|
eqeq12d |
|- ( a = M -> ( prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) <-> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
10 |
9
|
imbi2d |
|- ( a = M -> ( ( ph -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) ) <-> ( ph -> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
11 |
|
oveq2 |
|- ( a = n -> ( M ... a ) = ( M ... n ) ) |
12 |
11
|
prodeq1d |
|- ( a = n -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
13 |
11
|
sumeq1d |
|- ( a = n -> sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) |
14 |
13
|
fveq2d |
|- ( a = n -> ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) |
15 |
12 14
|
eqeq12d |
|- ( a = n -> ( prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) <-> prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
16 |
15
|
imbi2d |
|- ( a = n -> ( ( ph -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) ) <-> ( ph -> prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
17 |
|
oveq2 |
|- ( a = ( n + 1 ) -> ( M ... a ) = ( M ... ( n + 1 ) ) ) |
18 |
17
|
prodeq1d |
|- ( a = ( n + 1 ) -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
19 |
17
|
sumeq1d |
|- ( a = ( n + 1 ) -> sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) |
20 |
19
|
fveq2d |
|- ( a = ( n + 1 ) -> ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) |
21 |
18 20
|
eqeq12d |
|- ( a = ( n + 1 ) -> ( prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) <-> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
22 |
21
|
imbi2d |
|- ( a = ( n + 1 ) -> ( ( ph -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) ) <-> ( ph -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
23 |
|
oveq2 |
|- ( a = N -> ( M ... a ) = ( M ... N ) ) |
24 |
23
|
prodeq1d |
|- ( a = N -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
25 |
23
|
sumeq1d |
|- ( a = N -> sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) |
26 |
25
|
fveq2d |
|- ( a = N -> ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) ) |
27 |
24 26
|
eqeq12d |
|- ( a = N -> ( prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) <-> prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
28 |
27
|
imbi2d |
|- ( a = N -> ( ( ph -> prod_ m e. ( M ... a ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... a ) ( ( k e. Z |-> A ) ` m ) ) ) <-> ( ph -> prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
29 |
|
fzsn |
|- ( M e. ZZ -> ( M ... M ) = { M } ) |
30 |
29
|
adantl |
|- ( ( ph /\ M e. ZZ ) -> ( M ... M ) = { M } ) |
31 |
30
|
prodeq1d |
|- ( ( ph /\ M e. ZZ ) -> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. { M } ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
32 |
|
simpr |
|- ( ( ph /\ M e. ZZ ) -> M e. ZZ ) |
33 |
|
uzid |
|- ( M e. ZZ -> M e. ( ZZ>= ` M ) ) |
34 |
33 1
|
eleqtrrdi |
|- ( M e. ZZ -> M e. Z ) |
35 |
|
efcl |
|- ( A e. CC -> ( exp ` A ) e. CC ) |
36 |
3 35
|
syl |
|- ( ( ph /\ k e. Z ) -> ( exp ` A ) e. CC ) |
37 |
36
|
fmpttd |
|- ( ph -> ( k e. Z |-> ( exp ` A ) ) : Z --> CC ) |
38 |
37
|
ffvelrnda |
|- ( ( ph /\ M e. Z ) -> ( ( k e. Z |-> ( exp ` A ) ) ` M ) e. CC ) |
39 |
34 38
|
sylan2 |
|- ( ( ph /\ M e. ZZ ) -> ( ( k e. Z |-> ( exp ` A ) ) ` M ) e. CC ) |
40 |
|
fveq2 |
|- ( m = M -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( ( k e. Z |-> ( exp ` A ) ) ` M ) ) |
41 |
40
|
prodsn |
|- ( ( M e. ZZ /\ ( ( k e. Z |-> ( exp ` A ) ) ` M ) e. CC ) -> prod_ m e. { M } ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( ( k e. Z |-> ( exp ` A ) ) ` M ) ) |
42 |
32 39 41
|
syl2anc |
|- ( ( ph /\ M e. ZZ ) -> prod_ m e. { M } ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( ( k e. Z |-> ( exp ` A ) ) ` M ) ) |
43 |
34
|
adantl |
|- ( ( ph /\ M e. ZZ ) -> M e. Z ) |
44 |
|
fvex |
|- ( exp ` [_ M / k ]_ A ) e. _V |
45 |
|
nfcv |
|- F/_ k M |
46 |
|
nfcv |
|- F/_ k exp |
47 |
|
nfcsb1v |
|- F/_ k [_ M / k ]_ A |
48 |
46 47
|
nffv |
|- F/_ k ( exp ` [_ M / k ]_ A ) |
49 |
|
csbeq1a |
|- ( k = M -> A = [_ M / k ]_ A ) |
50 |
49
|
fveq2d |
|- ( k = M -> ( exp ` A ) = ( exp ` [_ M / k ]_ A ) ) |
51 |
|
eqid |
|- ( k e. Z |-> ( exp ` A ) ) = ( k e. Z |-> ( exp ` A ) ) |
52 |
45 48 50 51
|
fvmptf |
|- ( ( M e. Z /\ ( exp ` [_ M / k ]_ A ) e. _V ) -> ( ( k e. Z |-> ( exp ` A ) ) ` M ) = ( exp ` [_ M / k ]_ A ) ) |
53 |
43 44 52
|
sylancl |
|- ( ( ph /\ M e. ZZ ) -> ( ( k e. Z |-> ( exp ` A ) ) ` M ) = ( exp ` [_ M / k ]_ A ) ) |
54 |
31 42 53
|
3eqtrd |
|- ( ( ph /\ M e. ZZ ) -> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` [_ M / k ]_ A ) ) |
55 |
30
|
sumeq1d |
|- ( ( ph /\ M e. ZZ ) -> sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. { M } ( ( k e. Z |-> A ) ` m ) ) |
56 |
3
|
fmpttd |
|- ( ph -> ( k e. Z |-> A ) : Z --> CC ) |
57 |
56
|
ffvelrnda |
|- ( ( ph /\ M e. Z ) -> ( ( k e. Z |-> A ) ` M ) e. CC ) |
58 |
34 57
|
sylan2 |
|- ( ( ph /\ M e. ZZ ) -> ( ( k e. Z |-> A ) ` M ) e. CC ) |
59 |
|
fveq2 |
|- ( m = M -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` M ) ) |
60 |
59
|
sumsn |
|- ( ( M e. ZZ /\ ( ( k e. Z |-> A ) ` M ) e. CC ) -> sum_ m e. { M } ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` M ) ) |
61 |
32 58 60
|
syl2anc |
|- ( ( ph /\ M e. ZZ ) -> sum_ m e. { M } ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` M ) ) |
62 |
3
|
ralrimiva |
|- ( ph -> A. k e. Z A e. CC ) |
63 |
47
|
nfel1 |
|- F/ k [_ M / k ]_ A e. CC |
64 |
49
|
eleq1d |
|- ( k = M -> ( A e. CC <-> [_ M / k ]_ A e. CC ) ) |
65 |
63 64
|
rspc |
|- ( M e. Z -> ( A. k e. Z A e. CC -> [_ M / k ]_ A e. CC ) ) |
66 |
65
|
impcom |
|- ( ( A. k e. Z A e. CC /\ M e. Z ) -> [_ M / k ]_ A e. CC ) |
67 |
62 34 66
|
syl2an |
|- ( ( ph /\ M e. ZZ ) -> [_ M / k ]_ A e. CC ) |
68 |
|
eqid |
|- ( k e. Z |-> A ) = ( k e. Z |-> A ) |
69 |
68
|
fvmpts |
|- ( ( M e. Z /\ [_ M / k ]_ A e. CC ) -> ( ( k e. Z |-> A ) ` M ) = [_ M / k ]_ A ) |
70 |
43 67 69
|
syl2anc |
|- ( ( ph /\ M e. ZZ ) -> ( ( k e. Z |-> A ) ` M ) = [_ M / k ]_ A ) |
71 |
55 61 70
|
3eqtrd |
|- ( ( ph /\ M e. ZZ ) -> sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) = [_ M / k ]_ A ) |
72 |
71
|
fveq2d |
|- ( ( ph /\ M e. ZZ ) -> ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` [_ M / k ]_ A ) ) |
73 |
54 72
|
eqtr4d |
|- ( ( ph /\ M e. ZZ ) -> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) ) |
74 |
73
|
expcom |
|- ( M e. ZZ -> ( ph -> prod_ m e. ( M ... M ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... M ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
75 |
|
simp3 |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) |
76 |
1
|
peano2uzs |
|- ( n e. Z -> ( n + 1 ) e. Z ) |
77 |
|
simpr |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( n + 1 ) e. Z ) |
78 |
|
nfcsb1v |
|- F/_ k [_ ( n + 1 ) / k ]_ A |
79 |
78
|
nfel1 |
|- F/ k [_ ( n + 1 ) / k ]_ A e. CC |
80 |
|
csbeq1a |
|- ( k = ( n + 1 ) -> A = [_ ( n + 1 ) / k ]_ A ) |
81 |
80
|
eleq1d |
|- ( k = ( n + 1 ) -> ( A e. CC <-> [_ ( n + 1 ) / k ]_ A e. CC ) ) |
82 |
79 81
|
rspc |
|- ( ( n + 1 ) e. Z -> ( A. k e. Z A e. CC -> [_ ( n + 1 ) / k ]_ A e. CC ) ) |
83 |
62 82
|
mpan9 |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> [_ ( n + 1 ) / k ]_ A e. CC ) |
84 |
|
efcl |
|- ( [_ ( n + 1 ) / k ]_ A e. CC -> ( exp ` [_ ( n + 1 ) / k ]_ A ) e. CC ) |
85 |
83 84
|
syl |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( exp ` [_ ( n + 1 ) / k ]_ A ) e. CC ) |
86 |
|
nfcv |
|- F/_ k ( n + 1 ) |
87 |
46 78
|
nffv |
|- F/_ k ( exp ` [_ ( n + 1 ) / k ]_ A ) |
88 |
80
|
fveq2d |
|- ( k = ( n + 1 ) -> ( exp ` A ) = ( exp ` [_ ( n + 1 ) / k ]_ A ) ) |
89 |
86 87 88 51
|
fvmptf |
|- ( ( ( n + 1 ) e. Z /\ ( exp ` [_ ( n + 1 ) / k ]_ A ) e. CC ) -> ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) = ( exp ` [_ ( n + 1 ) / k ]_ A ) ) |
90 |
77 85 89
|
syl2anc |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) = ( exp ` [_ ( n + 1 ) / k ]_ A ) ) |
91 |
68
|
fvmpts |
|- ( ( ( n + 1 ) e. Z /\ [_ ( n + 1 ) / k ]_ A e. CC ) -> ( ( k e. Z |-> A ) ` ( n + 1 ) ) = [_ ( n + 1 ) / k ]_ A ) |
92 |
77 83 91
|
syl2anc |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( ( k e. Z |-> A ) ` ( n + 1 ) ) = [_ ( n + 1 ) / k ]_ A ) |
93 |
92
|
fveq2d |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) = ( exp ` [_ ( n + 1 ) / k ]_ A ) ) |
94 |
90 93
|
eqtr4d |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) = ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) |
95 |
76 94
|
sylan2 |
|- ( ( ph /\ n e. Z ) -> ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) = ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) |
96 |
95
|
3adant3 |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) = ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) |
97 |
75 96
|
oveq12d |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> ( prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) x. ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) ) = ( ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) x. ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
98 |
|
simpr |
|- ( ( ph /\ n e. Z ) -> n e. Z ) |
99 |
98 1
|
eleqtrdi |
|- ( ( ph /\ n e. Z ) -> n e. ( ZZ>= ` M ) ) |
100 |
|
elfzuz |
|- ( m e. ( M ... ( n + 1 ) ) -> m e. ( ZZ>= ` M ) ) |
101 |
100 1
|
eleqtrrdi |
|- ( m e. ( M ... ( n + 1 ) ) -> m e. Z ) |
102 |
37
|
ffvelrnda |
|- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) e. CC ) |
103 |
101 102
|
sylan2 |
|- ( ( ph /\ m e. ( M ... ( n + 1 ) ) ) -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) e. CC ) |
104 |
103
|
adantlr |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( M ... ( n + 1 ) ) ) -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) e. CC ) |
105 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) ) |
106 |
99 104 105
|
fprodp1 |
|- ( ( ph /\ n e. Z ) -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) x. ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) ) ) |
107 |
106
|
3adant3 |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) x. ( ( k e. Z |-> ( exp ` A ) ) ` ( n + 1 ) ) ) ) |
108 |
56
|
ffvelrnda |
|- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC ) |
109 |
101 108
|
sylan2 |
|- ( ( ph /\ m e. ( M ... ( n + 1 ) ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC ) |
110 |
109
|
adantlr |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( M ... ( n + 1 ) ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC ) |
111 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) |
112 |
99 110 111
|
fsump1 |
|- ( ( ph /\ n e. Z ) -> sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) = ( sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) + ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) |
113 |
112
|
fveq2d |
|- ( ( ph /\ n e. Z ) -> ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` ( sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) + ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
114 |
|
fzfid |
|- ( ( ph /\ n e. Z ) -> ( M ... n ) e. Fin ) |
115 |
|
elfzuz |
|- ( m e. ( M ... n ) -> m e. ( ZZ>= ` M ) ) |
116 |
115 1
|
eleqtrrdi |
|- ( m e. ( M ... n ) -> m e. Z ) |
117 |
116 108
|
sylan2 |
|- ( ( ph /\ m e. ( M ... n ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC ) |
118 |
117
|
adantlr |
|- ( ( ( ph /\ n e. Z ) /\ m e. ( M ... n ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC ) |
119 |
114 118
|
fsumcl |
|- ( ( ph /\ n e. Z ) -> sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) e. CC ) |
120 |
56
|
ffvelrnda |
|- ( ( ph /\ ( n + 1 ) e. Z ) -> ( ( k e. Z |-> A ) ` ( n + 1 ) ) e. CC ) |
121 |
76 120
|
sylan2 |
|- ( ( ph /\ n e. Z ) -> ( ( k e. Z |-> A ) ` ( n + 1 ) ) e. CC ) |
122 |
|
efadd |
|- ( ( sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) e. CC /\ ( ( k e. Z |-> A ) ` ( n + 1 ) ) e. CC ) -> ( exp ` ( sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) + ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) = ( ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) x. ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
123 |
119 121 122
|
syl2anc |
|- ( ( ph /\ n e. Z ) -> ( exp ` ( sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) + ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) = ( ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) x. ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
124 |
113 123
|
eqtrd |
|- ( ( ph /\ n e. Z ) -> ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) = ( ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) x. ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
125 |
124
|
3adant3 |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) = ( ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) x. ( exp ` ( ( k e. Z |-> A ) ` ( n + 1 ) ) ) ) ) |
126 |
97 107 125
|
3eqtr4d |
|- ( ( ph /\ n e. Z /\ prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) |
127 |
126
|
3exp |
|- ( ph -> ( n e. Z -> ( prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
128 |
127
|
com12 |
|- ( n e. Z -> ( ph -> ( prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
129 |
128
|
a2d |
|- ( n e. Z -> ( ( ph -> prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> ( ph -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
130 |
1
|
eqcomi |
|- ( ZZ>= ` M ) = Z |
131 |
129 130
|
eleq2s |
|- ( n e. ( ZZ>= ` M ) -> ( ( ph -> prod_ m e. ( M ... n ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... n ) ( ( k e. Z |-> A ) ` m ) ) ) -> ( ph -> prod_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... ( n + 1 ) ) ( ( k e. Z |-> A ) ` m ) ) ) ) ) |
132 |
10 16 22 28 74 131
|
uzind4 |
|- ( N e. ( ZZ>= ` M ) -> ( ph -> prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) ) ) |
133 |
4 132
|
mpcom |
|- ( ph -> prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) ) |
134 |
|
fvres |
|- ( m e. ( M ... N ) -> ( ( ( k e. Z |-> ( exp ` A ) ) |` ( M ... N ) ) ` m ) = ( ( k e. Z |-> ( exp ` A ) ) ` m ) ) |
135 |
|
fzssuz |
|- ( M ... N ) C_ ( ZZ>= ` M ) |
136 |
135 1
|
sseqtrri |
|- ( M ... N ) C_ Z |
137 |
|
resmpt |
|- ( ( M ... N ) C_ Z -> ( ( k e. Z |-> ( exp ` A ) ) |` ( M ... N ) ) = ( k e. ( M ... N ) |-> ( exp ` A ) ) ) |
138 |
136 137
|
ax-mp |
|- ( ( k e. Z |-> ( exp ` A ) ) |` ( M ... N ) ) = ( k e. ( M ... N ) |-> ( exp ` A ) ) |
139 |
138
|
fveq1i |
|- ( ( ( k e. Z |-> ( exp ` A ) ) |` ( M ... N ) ) ` m ) = ( ( k e. ( M ... N ) |-> ( exp ` A ) ) ` m ) |
140 |
134 139
|
eqtr3di |
|- ( m e. ( M ... N ) -> ( ( k e. Z |-> ( exp ` A ) ) ` m ) = ( ( k e. ( M ... N ) |-> ( exp ` A ) ) ` m ) ) |
141 |
140
|
prodeq2i |
|- prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ m e. ( M ... N ) ( ( k e. ( M ... N ) |-> ( exp ` A ) ) ` m ) |
142 |
|
prodfc |
|- prod_ m e. ( M ... N ) ( ( k e. ( M ... N ) |-> ( exp ` A ) ) ` m ) = prod_ k e. ( M ... N ) ( exp ` A ) |
143 |
141 142
|
eqtri |
|- prod_ m e. ( M ... N ) ( ( k e. Z |-> ( exp ` A ) ) ` m ) = prod_ k e. ( M ... N ) ( exp ` A ) |
144 |
|
fvres |
|- ( m e. ( M ... N ) -> ( ( ( k e. Z |-> A ) |` ( M ... N ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) ) |
145 |
|
resmpt |
|- ( ( M ... N ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... N ) ) = ( k e. ( M ... N ) |-> A ) ) |
146 |
136 145
|
ax-mp |
|- ( ( k e. Z |-> A ) |` ( M ... N ) ) = ( k e. ( M ... N ) |-> A ) |
147 |
146
|
fveq1i |
|- ( ( ( k e. Z |-> A ) |` ( M ... N ) ) ` m ) = ( ( k e. ( M ... N ) |-> A ) ` m ) |
148 |
144 147
|
eqtr3di |
|- ( m e. ( M ... N ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... N ) |-> A ) ` m ) ) |
149 |
148
|
sumeq2i |
|- sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... N ) ( ( k e. ( M ... N ) |-> A ) ` m ) |
150 |
|
sumfc |
|- sum_ m e. ( M ... N ) ( ( k e. ( M ... N ) |-> A ) ` m ) = sum_ k e. ( M ... N ) A |
151 |
149 150
|
eqtri |
|- sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) = sum_ k e. ( M ... N ) A |
152 |
151
|
fveq2i |
|- ( exp ` sum_ m e. ( M ... N ) ( ( k e. Z |-> A ) ` m ) ) = ( exp ` sum_ k e. ( M ... N ) A ) |
153 |
133 143 152
|
3eqtr3g |
|- ( ph -> prod_ k e. ( M ... N ) ( exp ` A ) = ( exp ` sum_ k e. ( M ... N ) A ) ) |