Step |
Hyp |
Ref |
Expression |
1 |
|
prodfn0.1 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
2 |
|
prodfn0.2 |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC ) |
3 |
|
prodfn0.3 |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) =/= 0 ) |
4 |
|
prodfrec.4 |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) = ( 1 / ( F ` k ) ) ) |
5 |
|
eluzfz2 |
|- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) ) |
6 |
1 5
|
syl |
|- ( ph -> N e. ( M ... N ) ) |
7 |
|
fveq2 |
|- ( m = M -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` M ) ) |
8 |
|
fveq2 |
|- ( m = M -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` M ) ) |
9 |
8
|
oveq2d |
|- ( m = M -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) |
10 |
7 9
|
eqeq12d |
|- ( m = M -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) ) |
11 |
10
|
imbi2d |
|- ( m = M -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) ) ) |
12 |
|
fveq2 |
|- ( m = n -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` n ) ) |
13 |
|
fveq2 |
|- ( m = n -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` n ) ) |
14 |
13
|
oveq2d |
|- ( m = n -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) |
15 |
12 14
|
eqeq12d |
|- ( m = n -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) ) |
16 |
15
|
imbi2d |
|- ( m = n -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) ) ) |
17 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` ( n + 1 ) ) ) |
18 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` ( n + 1 ) ) ) |
19 |
18
|
oveq2d |
|- ( m = ( n + 1 ) -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) |
20 |
17 19
|
eqeq12d |
|- ( m = ( n + 1 ) -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) |
21 |
20
|
imbi2d |
|- ( m = ( n + 1 ) -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) ) |
22 |
|
fveq2 |
|- ( m = N -> ( seq M ( x. , G ) ` m ) = ( seq M ( x. , G ) ` N ) ) |
23 |
|
fveq2 |
|- ( m = N -> ( seq M ( x. , F ) ` m ) = ( seq M ( x. , F ) ` N ) ) |
24 |
23
|
oveq2d |
|- ( m = N -> ( 1 / ( seq M ( x. , F ) ` m ) ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) |
25 |
22 24
|
eqeq12d |
|- ( m = N -> ( ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) <-> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) ) |
26 |
25
|
imbi2d |
|- ( m = N -> ( ( ph -> ( seq M ( x. , G ) ` m ) = ( 1 / ( seq M ( x. , F ) ` m ) ) ) <-> ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) ) ) |
27 |
|
eluzfz1 |
|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) ) |
28 |
1 27
|
syl |
|- ( ph -> M e. ( M ... N ) ) |
29 |
|
fveq2 |
|- ( k = M -> ( G ` k ) = ( G ` M ) ) |
30 |
|
fveq2 |
|- ( k = M -> ( F ` k ) = ( F ` M ) ) |
31 |
30
|
oveq2d |
|- ( k = M -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` M ) ) ) |
32 |
29 31
|
eqeq12d |
|- ( k = M -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` M ) = ( 1 / ( F ` M ) ) ) ) |
33 |
32
|
imbi2d |
|- ( k = M -> ( ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) <-> ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) ) ) |
34 |
4
|
expcom |
|- ( k e. ( M ... N ) -> ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) ) |
35 |
33 34
|
vtoclga |
|- ( M e. ( M ... N ) -> ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) ) |
36 |
28 35
|
mpcom |
|- ( ph -> ( G ` M ) = ( 1 / ( F ` M ) ) ) |
37 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
38 |
1 37
|
syl |
|- ( ph -> M e. ZZ ) |
39 |
|
seq1 |
|- ( M e. ZZ -> ( seq M ( x. , G ) ` M ) = ( G ` M ) ) |
40 |
38 39
|
syl |
|- ( ph -> ( seq M ( x. , G ) ` M ) = ( G ` M ) ) |
41 |
|
seq1 |
|- ( M e. ZZ -> ( seq M ( x. , F ) ` M ) = ( F ` M ) ) |
42 |
38 41
|
syl |
|- ( ph -> ( seq M ( x. , F ) ` M ) = ( F ` M ) ) |
43 |
42
|
oveq2d |
|- ( ph -> ( 1 / ( seq M ( x. , F ) ` M ) ) = ( 1 / ( F ` M ) ) ) |
44 |
36 40 43
|
3eqtr4d |
|- ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) |
45 |
44
|
a1i |
|- ( N e. ( ZZ>= ` M ) -> ( ph -> ( seq M ( x. , G ) ` M ) = ( 1 / ( seq M ( x. , F ) ` M ) ) ) ) |
46 |
|
oveq1 |
|- ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) ) |
47 |
46
|
3ad2ant3 |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) ) |
48 |
|
fzofzp1 |
|- ( n e. ( M ..^ N ) -> ( n + 1 ) e. ( M ... N ) ) |
49 |
|
fveq2 |
|- ( k = ( n + 1 ) -> ( G ` k ) = ( G ` ( n + 1 ) ) ) |
50 |
|
fveq2 |
|- ( k = ( n + 1 ) -> ( F ` k ) = ( F ` ( n + 1 ) ) ) |
51 |
50
|
oveq2d |
|- ( k = ( n + 1 ) -> ( 1 / ( F ` k ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) |
52 |
49 51
|
eqeq12d |
|- ( k = ( n + 1 ) -> ( ( G ` k ) = ( 1 / ( F ` k ) ) <-> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) |
53 |
52
|
imbi2d |
|- ( k = ( n + 1 ) -> ( ( ph -> ( G ` k ) = ( 1 / ( F ` k ) ) ) <-> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) ) |
54 |
53 34
|
vtoclga |
|- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) |
55 |
48 54
|
syl |
|- ( n e. ( M ..^ N ) -> ( ph -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) ) |
56 |
55
|
impcom |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( G ` ( n + 1 ) ) = ( 1 / ( F ` ( n + 1 ) ) ) ) |
57 |
56
|
oveq2d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) ) |
58 |
|
1cnd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> 1 e. CC ) |
59 |
|
elfzouz |
|- ( n e. ( M ..^ N ) -> n e. ( ZZ>= ` M ) ) |
60 |
59
|
adantl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> n e. ( ZZ>= ` M ) ) |
61 |
|
elfzouz2 |
|- ( n e. ( M ..^ N ) -> N e. ( ZZ>= ` n ) ) |
62 |
|
fzss2 |
|- ( N e. ( ZZ>= ` n ) -> ( M ... n ) C_ ( M ... N ) ) |
63 |
61 62
|
syl |
|- ( n e. ( M ..^ N ) -> ( M ... n ) C_ ( M ... N ) ) |
64 |
63
|
sselda |
|- ( ( n e. ( M ..^ N ) /\ k e. ( M ... n ) ) -> k e. ( M ... N ) ) |
65 |
64 2
|
sylan2 |
|- ( ( ph /\ ( n e. ( M ..^ N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) e. CC ) |
66 |
65
|
anassrs |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) e. CC ) |
67 |
|
mulcl |
|- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC ) |
68 |
67
|
adantl |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC ) |
69 |
60 66 68
|
seqcl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( seq M ( x. , F ) ` n ) e. CC ) |
70 |
50
|
eleq1d |
|- ( k = ( n + 1 ) -> ( ( F ` k ) e. CC <-> ( F ` ( n + 1 ) ) e. CC ) ) |
71 |
70
|
imbi2d |
|- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) e. CC ) <-> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) ) |
72 |
2
|
expcom |
|- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) e. CC ) ) |
73 |
71 72
|
vtoclga |
|- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) |
74 |
48 73
|
syl |
|- ( n e. ( M ..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) e. CC ) ) |
75 |
74
|
impcom |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( F ` ( n + 1 ) ) e. CC ) |
76 |
64 3
|
sylan2 |
|- ( ( ph /\ ( n e. ( M ..^ N ) /\ k e. ( M ... n ) ) ) -> ( F ` k ) =/= 0 ) |
77 |
76
|
anassrs |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ k e. ( M ... n ) ) -> ( F ` k ) =/= 0 ) |
78 |
60 66 77
|
prodfn0 |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( seq M ( x. , F ) ` n ) =/= 0 ) |
79 |
50
|
neeq1d |
|- ( k = ( n + 1 ) -> ( ( F ` k ) =/= 0 <-> ( F ` ( n + 1 ) ) =/= 0 ) ) |
80 |
79
|
imbi2d |
|- ( k = ( n + 1 ) -> ( ( ph -> ( F ` k ) =/= 0 ) <-> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) ) ) |
81 |
3
|
expcom |
|- ( k e. ( M ... N ) -> ( ph -> ( F ` k ) =/= 0 ) ) |
82 |
80 81
|
vtoclga |
|- ( ( n + 1 ) e. ( M ... N ) -> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) ) |
83 |
48 82
|
syl |
|- ( n e. ( M ..^ N ) -> ( ph -> ( F ` ( n + 1 ) ) =/= 0 ) ) |
84 |
83
|
impcom |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( F ` ( n + 1 ) ) =/= 0 ) |
85 |
58 69 58 75 78 84
|
divmuldivd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) = ( ( 1 x. 1 ) / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
86 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
87 |
86
|
oveq1i |
|- ( ( 1 x. 1 ) / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) |
88 |
85 87
|
eqtrdi |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( 1 / ( F ` ( n + 1 ) ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
89 |
57 88
|
eqtrd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
90 |
89
|
3adant3 |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( 1 / ( seq M ( x. , F ) ` n ) ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
91 |
47 90
|
eqtrd |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
92 |
|
seqp1 |
|- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) ) |
93 |
59 92
|
syl |
|- ( n e. ( M ..^ N ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) ) |
94 |
93
|
3ad2ant2 |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( ( seq M ( x. , G ) ` n ) x. ( G ` ( n + 1 ) ) ) ) |
95 |
|
seqp1 |
|- ( n e. ( ZZ>= ` M ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) |
96 |
59 95
|
syl |
|- ( n e. ( M ..^ N ) -> ( seq M ( x. , F ) ` ( n + 1 ) ) = ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) |
97 |
96
|
oveq2d |
|- ( n e. ( M ..^ N ) -> ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
98 |
97
|
3ad2ant2 |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) = ( 1 / ( ( seq M ( x. , F ) ` n ) x. ( F ` ( n + 1 ) ) ) ) ) |
99 |
91 94 98
|
3eqtr4d |
|- ( ( ph /\ n e. ( M ..^ N ) /\ ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) |
100 |
99
|
3exp |
|- ( ph -> ( n e. ( M ..^ N ) -> ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) ) |
101 |
100
|
com12 |
|- ( n e. ( M ..^ N ) -> ( ph -> ( ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) ) |
102 |
101
|
a2d |
|- ( n e. ( M ..^ N ) -> ( ( ph -> ( seq M ( x. , G ) ` n ) = ( 1 / ( seq M ( x. , F ) ` n ) ) ) -> ( ph -> ( seq M ( x. , G ) ` ( n + 1 ) ) = ( 1 / ( seq M ( x. , F ) ` ( n + 1 ) ) ) ) ) ) |
103 |
11 16 21 26 45 102
|
fzind2 |
|- ( N e. ( M ... N ) -> ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) ) |
104 |
6 103
|
mpcom |
|- ( ph -> ( seq M ( x. , G ) ` N ) = ( 1 / ( seq M ( x. , F ) ` N ) ) ) |