Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( n = N -> ( 1 ... n ) = ( 1 ... N ) ) |
2 |
1
|
sumeq1d |
|- ( n = N -> sum_ m e. ( 1 ... n ) ( 1 / m ) = sum_ m e. ( 1 ... N ) ( 1 / m ) ) |
3 |
|
fvoveq1 |
|- ( n = N -> ( log ` ( n + 1 ) ) = ( log ` ( N + 1 ) ) ) |
4 |
2 3
|
oveq12d |
|- ( n = N -> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) = ( sum_ m e. ( 1 ... N ) ( 1 / m ) - ( log ` ( N + 1 ) ) ) ) |
5 |
4
|
eleq1d |
|- ( n = N -> ( ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) <-> ( sum_ m e. ( 1 ... N ) ( 1 / m ) - ( log ` ( N + 1 ) ) ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) ) |
6 |
|
eqid |
|- ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) ) = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) ) |
7 |
|
eqid |
|- ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) ) = ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) ) |
8 |
|
eqid |
|- ( n e. NN |-> ( log ` ( 1 + ( 1 / n ) ) ) ) = ( n e. NN |-> ( log ` ( 1 + ( 1 / n ) ) ) ) |
9 |
|
oveq2 |
|- ( k = n -> ( 1 / k ) = ( 1 / n ) ) |
10 |
9
|
oveq2d |
|- ( k = n -> ( 1 + ( 1 / k ) ) = ( 1 + ( 1 / n ) ) ) |
11 |
10
|
fveq2d |
|- ( k = n -> ( log ` ( 1 + ( 1 / k ) ) ) = ( log ` ( 1 + ( 1 / n ) ) ) ) |
12 |
9 11
|
oveq12d |
|- ( k = n -> ( ( 1 / k ) - ( log ` ( 1 + ( 1 / k ) ) ) ) = ( ( 1 / n ) - ( log ` ( 1 + ( 1 / n ) ) ) ) ) |
13 |
12
|
cbvmptv |
|- ( k e. NN |-> ( ( 1 / k ) - ( log ` ( 1 + ( 1 / k ) ) ) ) ) = ( n e. NN |-> ( ( 1 / n ) - ( log ` ( 1 + ( 1 / n ) ) ) ) ) |
14 |
6 7 8 13
|
emcllem7 |
|- ( gamma e. ( ( 1 - ( log ` 2 ) ) [,] 1 ) /\ ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` n ) ) ) : NN --> ( gamma [,] 1 ) /\ ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) ) : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) |
15 |
14
|
simp3i |
|- ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) ) : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) |
16 |
7
|
fmpt |
|- ( A. n e. NN ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) <-> ( n e. NN |-> ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) ) : NN --> ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) |
17 |
15 16
|
mpbir |
|- A. n e. NN ( sum_ m e. ( 1 ... n ) ( 1 / m ) - ( log ` ( n + 1 ) ) ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) |
18 |
5 17
|
vtoclri |
|- ( N e. NN -> ( sum_ m e. ( 1 ... N ) ( 1 / m ) - ( log ` ( N + 1 ) ) ) e. ( ( 1 - ( log ` 2 ) ) [,] gamma ) ) |