Step |
Hyp |
Ref |
Expression |
1 |
|
gamcvg2.f |
|- F = ( m e. NN |-> ( ( ( ( m + 1 ) / m ) ^c A ) / ( ( A / m ) + 1 ) ) ) |
2 |
|
gamcvg2.a |
|- ( ph -> A e. ( CC \ ( ZZ \ NN ) ) ) |
3 |
|
gamcvg2.g |
|- G = ( m e. NN |-> ( ( A x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( A / m ) + 1 ) ) ) ) |
4 |
|
addcl |
|- ( ( n e. CC /\ x e. CC ) -> ( n + x ) e. CC ) |
5 |
4
|
adantl |
|- ( ( ( ph /\ k e. NN ) /\ ( n e. CC /\ x e. CC ) ) -> ( n + x ) e. CC ) |
6 |
|
simpll |
|- ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ph ) |
7 |
|
elfznn |
|- ( n e. ( 1 ... k ) -> n e. NN ) |
8 |
7
|
adantl |
|- ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> n e. NN ) |
9 |
|
oveq1 |
|- ( m = n -> ( m + 1 ) = ( n + 1 ) ) |
10 |
|
id |
|- ( m = n -> m = n ) |
11 |
9 10
|
oveq12d |
|- ( m = n -> ( ( m + 1 ) / m ) = ( ( n + 1 ) / n ) ) |
12 |
11
|
fveq2d |
|- ( m = n -> ( log ` ( ( m + 1 ) / m ) ) = ( log ` ( ( n + 1 ) / n ) ) ) |
13 |
12
|
oveq2d |
|- ( m = n -> ( A x. ( log ` ( ( m + 1 ) / m ) ) ) = ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) |
14 |
|
oveq2 |
|- ( m = n -> ( A / m ) = ( A / n ) ) |
15 |
14
|
oveq1d |
|- ( m = n -> ( ( A / m ) + 1 ) = ( ( A / n ) + 1 ) ) |
16 |
15
|
fveq2d |
|- ( m = n -> ( log ` ( ( A / m ) + 1 ) ) = ( log ` ( ( A / n ) + 1 ) ) ) |
17 |
13 16
|
oveq12d |
|- ( m = n -> ( ( A x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( A / m ) + 1 ) ) ) = ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) |
18 |
|
ovex |
|- ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) e. _V |
19 |
17 3 18
|
fvmpt |
|- ( n e. NN -> ( G ` n ) = ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) |
20 |
19
|
adantl |
|- ( ( ph /\ n e. NN ) -> ( G ` n ) = ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) |
21 |
2
|
adantr |
|- ( ( ph /\ n e. NN ) -> A e. ( CC \ ( ZZ \ NN ) ) ) |
22 |
21
|
eldifad |
|- ( ( ph /\ n e. NN ) -> A e. CC ) |
23 |
|
simpr |
|- ( ( ph /\ n e. NN ) -> n e. NN ) |
24 |
23
|
peano2nnd |
|- ( ( ph /\ n e. NN ) -> ( n + 1 ) e. NN ) |
25 |
24
|
nnrpd |
|- ( ( ph /\ n e. NN ) -> ( n + 1 ) e. RR+ ) |
26 |
23
|
nnrpd |
|- ( ( ph /\ n e. NN ) -> n e. RR+ ) |
27 |
25 26
|
rpdivcld |
|- ( ( ph /\ n e. NN ) -> ( ( n + 1 ) / n ) e. RR+ ) |
28 |
27
|
relogcld |
|- ( ( ph /\ n e. NN ) -> ( log ` ( ( n + 1 ) / n ) ) e. RR ) |
29 |
28
|
recnd |
|- ( ( ph /\ n e. NN ) -> ( log ` ( ( n + 1 ) / n ) ) e. CC ) |
30 |
22 29
|
mulcld |
|- ( ( ph /\ n e. NN ) -> ( A x. ( log ` ( ( n + 1 ) / n ) ) ) e. CC ) |
31 |
23
|
nncnd |
|- ( ( ph /\ n e. NN ) -> n e. CC ) |
32 |
23
|
nnne0d |
|- ( ( ph /\ n e. NN ) -> n =/= 0 ) |
33 |
22 31 32
|
divcld |
|- ( ( ph /\ n e. NN ) -> ( A / n ) e. CC ) |
34 |
|
1cnd |
|- ( ( ph /\ n e. NN ) -> 1 e. CC ) |
35 |
33 34
|
addcld |
|- ( ( ph /\ n e. NN ) -> ( ( A / n ) + 1 ) e. CC ) |
36 |
21 23
|
dmgmdivn0 |
|- ( ( ph /\ n e. NN ) -> ( ( A / n ) + 1 ) =/= 0 ) |
37 |
35 36
|
logcld |
|- ( ( ph /\ n e. NN ) -> ( log ` ( ( A / n ) + 1 ) ) e. CC ) |
38 |
30 37
|
subcld |
|- ( ( ph /\ n e. NN ) -> ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) e. CC ) |
39 |
20 38
|
eqeltrd |
|- ( ( ph /\ n e. NN ) -> ( G ` n ) e. CC ) |
40 |
6 8 39
|
syl2anc |
|- ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( G ` n ) e. CC ) |
41 |
|
simpr |
|- ( ( ph /\ k e. NN ) -> k e. NN ) |
42 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
43 |
41 42
|
eleqtrdi |
|- ( ( ph /\ k e. NN ) -> k e. ( ZZ>= ` 1 ) ) |
44 |
|
efadd |
|- ( ( n e. CC /\ x e. CC ) -> ( exp ` ( n + x ) ) = ( ( exp ` n ) x. ( exp ` x ) ) ) |
45 |
44
|
adantl |
|- ( ( ( ph /\ k e. NN ) /\ ( n e. CC /\ x e. CC ) ) -> ( exp ` ( n + x ) ) = ( ( exp ` n ) x. ( exp ` x ) ) ) |
46 |
|
efsub |
|- ( ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) e. CC /\ ( log ` ( ( A / n ) + 1 ) ) e. CC ) -> ( exp ` ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) = ( ( exp ` ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) / ( exp ` ( log ` ( ( A / n ) + 1 ) ) ) ) ) |
47 |
30 37 46
|
syl2anc |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) = ( ( exp ` ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) / ( exp ` ( log ` ( ( A / n ) + 1 ) ) ) ) ) |
48 |
31 34
|
addcld |
|- ( ( ph /\ n e. NN ) -> ( n + 1 ) e. CC ) |
49 |
48 31 32
|
divcld |
|- ( ( ph /\ n e. NN ) -> ( ( n + 1 ) / n ) e. CC ) |
50 |
24
|
nnne0d |
|- ( ( ph /\ n e. NN ) -> ( n + 1 ) =/= 0 ) |
51 |
48 31 50 32
|
divne0d |
|- ( ( ph /\ n e. NN ) -> ( ( n + 1 ) / n ) =/= 0 ) |
52 |
49 51 22
|
cxpefd |
|- ( ( ph /\ n e. NN ) -> ( ( ( n + 1 ) / n ) ^c A ) = ( exp ` ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) ) |
53 |
52
|
eqcomd |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) = ( ( ( n + 1 ) / n ) ^c A ) ) |
54 |
|
eflog |
|- ( ( ( ( A / n ) + 1 ) e. CC /\ ( ( A / n ) + 1 ) =/= 0 ) -> ( exp ` ( log ` ( ( A / n ) + 1 ) ) ) = ( ( A / n ) + 1 ) ) |
55 |
35 36 54
|
syl2anc |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( log ` ( ( A / n ) + 1 ) ) ) = ( ( A / n ) + 1 ) ) |
56 |
53 55
|
oveq12d |
|- ( ( ph /\ n e. NN ) -> ( ( exp ` ( A x. ( log ` ( ( n + 1 ) / n ) ) ) ) / ( exp ` ( log ` ( ( A / n ) + 1 ) ) ) ) = ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) ) |
57 |
47 56
|
eqtrd |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) = ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) ) |
58 |
20
|
fveq2d |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( G ` n ) ) = ( exp ` ( ( A x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( A / n ) + 1 ) ) ) ) ) |
59 |
11
|
oveq1d |
|- ( m = n -> ( ( ( m + 1 ) / m ) ^c A ) = ( ( ( n + 1 ) / n ) ^c A ) ) |
60 |
59 15
|
oveq12d |
|- ( m = n -> ( ( ( ( m + 1 ) / m ) ^c A ) / ( ( A / m ) + 1 ) ) = ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) ) |
61 |
|
ovex |
|- ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) e. _V |
62 |
60 1 61
|
fvmpt |
|- ( n e. NN -> ( F ` n ) = ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) ) |
63 |
62
|
adantl |
|- ( ( ph /\ n e. NN ) -> ( F ` n ) = ( ( ( ( n + 1 ) / n ) ^c A ) / ( ( A / n ) + 1 ) ) ) |
64 |
57 58 63
|
3eqtr4d |
|- ( ( ph /\ n e. NN ) -> ( exp ` ( G ` n ) ) = ( F ` n ) ) |
65 |
6 8 64
|
syl2anc |
|- ( ( ( ph /\ k e. NN ) /\ n e. ( 1 ... k ) ) -> ( exp ` ( G ` n ) ) = ( F ` n ) ) |
66 |
5 40 43 45 65
|
seqhomo |
|- ( ( ph /\ k e. NN ) -> ( exp ` ( seq 1 ( + , G ) ` k ) ) = ( seq 1 ( x. , F ) ` k ) ) |
67 |
66
|
mpteq2dva |
|- ( ph -> ( k e. NN |-> ( exp ` ( seq 1 ( + , G ) ` k ) ) ) = ( k e. NN |-> ( seq 1 ( x. , F ) ` k ) ) ) |
68 |
|
eff |
|- exp : CC --> CC |
69 |
68
|
a1i |
|- ( ph -> exp : CC --> CC ) |
70 |
|
1z |
|- 1 e. ZZ |
71 |
70
|
a1i |
|- ( ph -> 1 e. ZZ ) |
72 |
42 71 39
|
serf |
|- ( ph -> seq 1 ( + , G ) : NN --> CC ) |
73 |
|
fcompt |
|- ( ( exp : CC --> CC /\ seq 1 ( + , G ) : NN --> CC ) -> ( exp o. seq 1 ( + , G ) ) = ( k e. NN |-> ( exp ` ( seq 1 ( + , G ) ` k ) ) ) ) |
74 |
69 72 73
|
syl2anc |
|- ( ph -> ( exp o. seq 1 ( + , G ) ) = ( k e. NN |-> ( exp ` ( seq 1 ( + , G ) ` k ) ) ) ) |
75 |
|
seqfn |
|- ( 1 e. ZZ -> seq 1 ( x. , F ) Fn ( ZZ>= ` 1 ) ) |
76 |
70 75
|
mp1i |
|- ( ph -> seq 1 ( x. , F ) Fn ( ZZ>= ` 1 ) ) |
77 |
42
|
fneq2i |
|- ( seq 1 ( x. , F ) Fn NN <-> seq 1 ( x. , F ) Fn ( ZZ>= ` 1 ) ) |
78 |
76 77
|
sylibr |
|- ( ph -> seq 1 ( x. , F ) Fn NN ) |
79 |
|
dffn5 |
|- ( seq 1 ( x. , F ) Fn NN <-> seq 1 ( x. , F ) = ( k e. NN |-> ( seq 1 ( x. , F ) ` k ) ) ) |
80 |
78 79
|
sylib |
|- ( ph -> seq 1 ( x. , F ) = ( k e. NN |-> ( seq 1 ( x. , F ) ` k ) ) ) |
81 |
67 74 80
|
3eqtr4d |
|- ( ph -> ( exp o. seq 1 ( + , G ) ) = seq 1 ( x. , F ) ) |