Metamath Proof Explorer


Theorem harmonicbnd2

Description: A bound on the harmonic series, as compared to the natural logarithm. (Contributed by Mario Carneiro, 13-Apr-2016)

Ref Expression
Assertion harmonicbnd2 N m = 1 N 1 m log N + 1 1 log 2 γ

Proof

Step Hyp Ref Expression
1 oveq2 n = N 1 n = 1 N
2 1 sumeq1d n = N m = 1 n 1 m = m = 1 N 1 m
3 fvoveq1 n = N log n + 1 = log N + 1
4 2 3 oveq12d n = N m = 1 n 1 m log n + 1 = m = 1 N 1 m log N + 1
5 4 eleq1d n = N m = 1 n 1 m log n + 1 1 log 2 γ m = 1 N 1 m log N + 1 1 log 2 γ
6 eqid n m = 1 n 1 m log n = n m = 1 n 1 m log n
7 eqid n m = 1 n 1 m log n + 1 = n m = 1 n 1 m log n + 1
8 eqid n log 1 + 1 n = n 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 1 k log 1 + 1 k = n 1 n log 1 + 1 n
14 6 7 8 13 emcllem7 γ 1 log 2 1 n m = 1 n 1 m log n : γ 1 n m = 1 n 1 m log n + 1 : 1 log 2 γ
15 14 simp3i n m = 1 n 1 m log n + 1 : 1 log 2 γ
16 7 fmpt n m = 1 n 1 m log n + 1 1 log 2 γ n m = 1 n 1 m log n + 1 : 1 log 2 γ
17 15 16 mpbir n m = 1 n 1 m log n + 1 1 log 2 γ
18 5 17 vtoclri N m = 1 N 1 m log N + 1 1 log 2 γ