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Theorem mulog2sumlem1 27037
Description: Asymptotic formula for Σ𝑛 ≀ π‘₯, log(π‘₯ / 𝑛) / 𝑛 = (1 / 2)log↑2(π‘₯) + Ξ³ Β· logπ‘₯ βˆ’ 𝐿 + 𝑂(logπ‘₯ / π‘₯), with explicit constants. Equation 10.2.7 of [Shapiro], p. 407. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
logdivsum.1 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘–) / 𝑖) βˆ’ (((logβ€˜π‘¦)↑2) / 2)))
mulog2sumlem.1 (πœ‘ β†’ 𝐹 β‡π‘Ÿ 𝐿)
mulog2sumlem1.2 (πœ‘ β†’ 𝐴 ∈ ℝ+)
mulog2sumlem1.3 (πœ‘ β†’ e ≀ 𝐴)
Assertion
Ref Expression
mulog2sumlem1 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))) ≀ (2 Β· ((logβ€˜π΄) / 𝐴)))
Distinct variable groups:   𝑖,π‘š,𝑦,𝐴   πœ‘,π‘š
Allowed substitution hints:   πœ‘(𝑦,𝑖)   𝐹(𝑦,𝑖,π‘š)   𝐿(𝑦,𝑖,π‘š)

Proof of Theorem mulog2sumlem1
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 fzfid 13938 . . . . . 6 (πœ‘ β†’ (1...(βŒŠβ€˜π΄)) ∈ Fin)
2 mulog2sumlem1.2 . . . . . . . . 9 (πœ‘ β†’ 𝐴 ∈ ℝ+)
3 elfznn 13530 . . . . . . . . . 10 (π‘š ∈ (1...(βŒŠβ€˜π΄)) β†’ π‘š ∈ β„•)
43nnrpd 13014 . . . . . . . . 9 (π‘š ∈ (1...(βŒŠβ€˜π΄)) β†’ π‘š ∈ ℝ+)
5 rpdivcl 12999 . . . . . . . . 9 ((𝐴 ∈ ℝ+ ∧ π‘š ∈ ℝ+) β†’ (𝐴 / π‘š) ∈ ℝ+)
62, 4, 5syl2an 597 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (𝐴 / π‘š) ∈ ℝ+)
76relogcld 26131 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (logβ€˜(𝐴 / π‘š)) ∈ ℝ)
83adantl 483 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ π‘š ∈ β„•)
97, 8nndivred 12266 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜(𝐴 / π‘š)) / π‘š) ∈ ℝ)
101, 9fsumrecl 15680 . . . . 5 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) ∈ ℝ)
112relogcld 26131 . . . . . . . 8 (πœ‘ β†’ (logβ€˜π΄) ∈ ℝ)
1211resqcld 14090 . . . . . . 7 (πœ‘ β†’ ((logβ€˜π΄)↑2) ∈ ℝ)
1312rehalfcld 12459 . . . . . 6 (πœ‘ β†’ (((logβ€˜π΄)↑2) / 2) ∈ ℝ)
14 emre 26510 . . . . . . . 8 Ξ³ ∈ ℝ
15 remulcl 11195 . . . . . . . 8 ((Ξ³ ∈ ℝ ∧ (logβ€˜π΄) ∈ ℝ) β†’ (Ξ³ Β· (logβ€˜π΄)) ∈ ℝ)
1614, 11, 15sylancr 588 . . . . . . 7 (πœ‘ β†’ (Ξ³ Β· (logβ€˜π΄)) ∈ ℝ)
17 rpsup 13831 . . . . . . . . 9 sup(ℝ+, ℝ*, < ) = +∞
1817a1i 11 . . . . . . . 8 (πœ‘ β†’ sup(ℝ+, ℝ*, < ) = +∞)
19 logdivsum.1 . . . . . . . . . . . . 13 𝐹 = (𝑦 ∈ ℝ+ ↦ (Σ𝑖 ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘–) / 𝑖) βˆ’ (((logβ€˜π‘¦)↑2) / 2)))
2019logdivsum 27036 . . . . . . . . . . . 12 (𝐹:ℝ+βŸΆβ„ ∧ 𝐹 ∈ dom β‡π‘Ÿ ∧ ((𝐹 β‡π‘Ÿ 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≀ 𝐴) β†’ (absβ€˜((πΉβ€˜π΄) βˆ’ 𝐿)) ≀ ((logβ€˜π΄) / 𝐴)))
2120simp1i 1140 . . . . . . . . . . 11 𝐹:ℝ+βŸΆβ„
2221a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 𝐹:ℝ+βŸΆβ„)
2322feqmptd 6961 . . . . . . . . 9 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ ℝ+ ↦ (πΉβ€˜π‘₯)))
24 mulog2sumlem.1 . . . . . . . . 9 (πœ‘ β†’ 𝐹 β‡π‘Ÿ 𝐿)
2523, 24eqbrtrrd 5173 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ ℝ+ ↦ (πΉβ€˜π‘₯)) β‡π‘Ÿ 𝐿)
2621ffvelcdmi 7086 . . . . . . . . 9 (π‘₯ ∈ ℝ+ β†’ (πΉβ€˜π‘₯) ∈ ℝ)
2726adantl 483 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ+) β†’ (πΉβ€˜π‘₯) ∈ ℝ)
2818, 25, 27rlimrecl 15524 . . . . . . 7 (πœ‘ β†’ 𝐿 ∈ ℝ)
2916, 28resubcld 11642 . . . . . 6 (πœ‘ β†’ ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿) ∈ ℝ)
3013, 29readdcld 11243 . . . . 5 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)) ∈ ℝ)
3110, 30resubcld 11642 . . . 4 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿))) ∈ ℝ)
3231recnd 11242 . . 3 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿))) ∈ β„‚)
3332abscld 15383 . 2 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))) ∈ ℝ)
34 rerpdivcl 13004 . . . . . . . 8 (((logβ€˜π΄) ∈ ℝ ∧ π‘š ∈ ℝ+) β†’ ((logβ€˜π΄) / π‘š) ∈ ℝ)
3511, 4, 34syl2an 597 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜π΄) / π‘š) ∈ ℝ)
3635recnd 11242 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜π΄) / π‘š) ∈ β„‚)
371, 36fsumcl 15679 . . . . 5 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) ∈ β„‚)
3811recnd 11242 . . . . . 6 (πœ‘ β†’ (logβ€˜π΄) ∈ β„‚)
39 readdcl 11193 . . . . . . . 8 (((logβ€˜π΄) ∈ ℝ ∧ Ξ³ ∈ ℝ) β†’ ((logβ€˜π΄) + Ξ³) ∈ ℝ)
4011, 14, 39sylancl 587 . . . . . . 7 (πœ‘ β†’ ((logβ€˜π΄) + Ξ³) ∈ ℝ)
4140recnd 11242 . . . . . 6 (πœ‘ β†’ ((logβ€˜π΄) + Ξ³) ∈ β„‚)
4238, 41mulcld 11234 . . . . 5 (πœ‘ β†’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) ∈ β„‚)
4337, 42subcld 11571 . . . 4 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) ∈ β„‚)
4443abscld 15383 . . 3 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) ∈ ℝ)
458nnrpd 13014 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ π‘š ∈ ℝ+)
4645relogcld 26131 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (logβ€˜π‘š) ∈ ℝ)
4746, 8nndivred 12266 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜π‘š) / π‘š) ∈ ℝ)
4847recnd 11242 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜π‘š) / π‘š) ∈ β„‚)
491, 48fsumcl 15679 . . . . 5 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) ∈ β„‚)
5013recnd 11242 . . . . . 6 (πœ‘ β†’ (((logβ€˜π΄)↑2) / 2) ∈ β„‚)
5128recnd 11242 . . . . . 6 (πœ‘ β†’ 𝐿 ∈ β„‚)
5250, 51addcld 11233 . . . . 5 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + 𝐿) ∈ β„‚)
5349, 52subcld 11571 . . . 4 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)) ∈ β„‚)
5453abscld 15383 . . 3 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))) ∈ ℝ)
5544, 54readdcld 11243 . 2 (πœ‘ β†’ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) + (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))) ∈ ℝ)
56 2re 12286 . . 3 2 ∈ ℝ
5711, 2rerpdivcld 13047 . . 3 (πœ‘ β†’ ((logβ€˜π΄) / 𝐴) ∈ ℝ)
58 remulcl 11195 . . 3 ((2 ∈ ℝ ∧ ((logβ€˜π΄) / 𝐴) ∈ ℝ) β†’ (2 Β· ((logβ€˜π΄) / 𝐴)) ∈ ℝ)
5956, 57, 58sylancr 588 . 2 (πœ‘ β†’ (2 Β· ((logβ€˜π΄) / 𝐴)) ∈ ℝ)
60 relogdiv 26101 . . . . . . . . . . 11 ((𝐴 ∈ ℝ+ ∧ π‘š ∈ ℝ+) β†’ (logβ€˜(𝐴 / π‘š)) = ((logβ€˜π΄) βˆ’ (logβ€˜π‘š)))
612, 4, 60syl2an 597 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (logβ€˜(𝐴 / π‘š)) = ((logβ€˜π΄) βˆ’ (logβ€˜π‘š)))
6261oveq1d 7424 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜(𝐴 / π‘š)) / π‘š) = (((logβ€˜π΄) βˆ’ (logβ€˜π‘š)) / π‘š))
6338adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (logβ€˜π΄) ∈ β„‚)
6446recnd 11242 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (logβ€˜π‘š) ∈ β„‚)
6545rpcnne0d 13025 . . . . . . . . . 10 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (π‘š ∈ β„‚ ∧ π‘š β‰  0))
66 divsubdir 11908 . . . . . . . . . 10 (((logβ€˜π΄) ∈ β„‚ ∧ (logβ€˜π‘š) ∈ β„‚ ∧ (π‘š ∈ β„‚ ∧ π‘š β‰  0)) β†’ (((logβ€˜π΄) βˆ’ (logβ€˜π‘š)) / π‘š) = (((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π‘š) / π‘š)))
6763, 64, 65, 66syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (((logβ€˜π΄) βˆ’ (logβ€˜π‘š)) / π‘š) = (((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π‘š) / π‘š)))
6862, 67eqtrd 2773 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜(𝐴 / π‘š)) / π‘š) = (((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π‘š) / π‘š)))
6968sumeq2dv 15649 . . . . . . 7 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π‘š) / π‘š)))
701, 36, 48fsumsub 15734 . . . . . . 7 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π‘š) / π‘š)) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š)))
7169, 70eqtrd 2773 . . . . . 6 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š)))
72 remulcl 11195 . . . . . . . . . . . . 13 (((logβ€˜π΄) ∈ ℝ ∧ Ξ³ ∈ ℝ) β†’ ((logβ€˜π΄) Β· Ξ³) ∈ ℝ)
7311, 14, 72sylancl 587 . . . . . . . . . . . 12 (πœ‘ β†’ ((logβ€˜π΄) Β· Ξ³) ∈ ℝ)
7413, 73readdcld 11243 . . . . . . . . . . 11 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) ∈ ℝ)
7574recnd 11242 . . . . . . . . . 10 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) ∈ β„‚)
7675, 50pncand 11572 . . . . . . . . 9 (πœ‘ β†’ ((((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) + (((logβ€˜π΄)↑2) / 2)) βˆ’ (((logβ€˜π΄)↑2) / 2)) = ((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)))
7714recni 11228 . . . . . . . . . . . . 13 Ξ³ ∈ β„‚
7877a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ Ξ³ ∈ β„‚)
7938, 38, 78adddid 11238 . . . . . . . . . . 11 (πœ‘ β†’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) = (((logβ€˜π΄) Β· (logβ€˜π΄)) + ((logβ€˜π΄) Β· Ξ³)))
8012recnd 11242 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((logβ€˜π΄)↑2) ∈ β„‚)
81802halvesd 12458 . . . . . . . . . . . . 13 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + (((logβ€˜π΄)↑2) / 2)) = ((logβ€˜π΄)↑2))
8238sqvald 14108 . . . . . . . . . . . . 13 (πœ‘ β†’ ((logβ€˜π΄)↑2) = ((logβ€˜π΄) Β· (logβ€˜π΄)))
8381, 82eqtrd 2773 . . . . . . . . . . . 12 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + (((logβ€˜π΄)↑2) / 2)) = ((logβ€˜π΄) Β· (logβ€˜π΄)))
8483oveq1d 7424 . . . . . . . . . . 11 (πœ‘ β†’ (((((logβ€˜π΄)↑2) / 2) + (((logβ€˜π΄)↑2) / 2)) + ((logβ€˜π΄) Β· Ξ³)) = (((logβ€˜π΄) Β· (logβ€˜π΄)) + ((logβ€˜π΄) Β· Ξ³)))
8573recnd 11242 . . . . . . . . . . . 12 (πœ‘ β†’ ((logβ€˜π΄) Β· Ξ³) ∈ β„‚)
8650, 50, 85add32d 11441 . . . . . . . . . . 11 (πœ‘ β†’ (((((logβ€˜π΄)↑2) / 2) + (((logβ€˜π΄)↑2) / 2)) + ((logβ€˜π΄) Β· Ξ³)) = (((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) + (((logβ€˜π΄)↑2) / 2)))
8779, 84, 863eqtr2d 2779 . . . . . . . . . 10 (πœ‘ β†’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) = (((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) + (((logβ€˜π΄)↑2) / 2)))
8887oveq1d 7424 . . . . . . . . 9 (πœ‘ β†’ (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ (((logβ€˜π΄)↑2) / 2)) = ((((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)) + (((logβ€˜π΄)↑2) / 2)) βˆ’ (((logβ€˜π΄)↑2) / 2)))
89 mulcom 11196 . . . . . . . . . . 11 ((Ξ³ ∈ β„‚ ∧ (logβ€˜π΄) ∈ β„‚) β†’ (Ξ³ Β· (logβ€˜π΄)) = ((logβ€˜π΄) Β· Ξ³))
9077, 38, 89sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ (Ξ³ Β· (logβ€˜π΄)) = ((logβ€˜π΄) Β· Ξ³))
9190oveq2d 7425 . . . . . . . . 9 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + (Ξ³ Β· (logβ€˜π΄))) = ((((logβ€˜π΄)↑2) / 2) + ((logβ€˜π΄) Β· Ξ³)))
9276, 88, 913eqtr4rd 2784 . . . . . . . 8 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + (Ξ³ Β· (logβ€˜π΄))) = (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ (((logβ€˜π΄)↑2) / 2)))
9392oveq1d 7424 . . . . . . 7 (πœ‘ β†’ (((((logβ€˜π΄)↑2) / 2) + (Ξ³ Β· (logβ€˜π΄))) βˆ’ 𝐿) = ((((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ (((logβ€˜π΄)↑2) / 2)) βˆ’ 𝐿))
9490, 85eqeltrd 2834 . . . . . . . 8 (πœ‘ β†’ (Ξ³ Β· (logβ€˜π΄)) ∈ β„‚)
9550, 94, 51addsubassd 11591 . . . . . . 7 (πœ‘ β†’ (((((logβ€˜π΄)↑2) / 2) + (Ξ³ Β· (logβ€˜π΄))) βˆ’ 𝐿) = ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))
9642, 50, 51subsub4d 11602 . . . . . . 7 (πœ‘ β†’ ((((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ (((logβ€˜π΄)↑2) / 2)) βˆ’ 𝐿) = (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))
9793, 95, 963eqtr3d 2781 . . . . . 6 (πœ‘ β†’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)) = (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))
9871, 97oveq12d 7427 . . . . 5 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿))) = ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š)) βˆ’ (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))))
9937, 49, 42, 52sub4d 11620 . . . . 5 (πœ‘ β†’ ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š)) βˆ’ (((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))) = ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) βˆ’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))))
10098, 99eqtrd 2773 . . . 4 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿))) = ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) βˆ’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))))
101100fveq2d 6896 . . 3 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))) = (absβ€˜((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) βˆ’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))))
10243, 53abs2dif2d 15405 . . 3 (πœ‘ β†’ (absβ€˜((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) βˆ’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))) ≀ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) + (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))))
103101, 102eqbrtrd 5171 . 2 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))) ≀ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) + (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))))
104 harmonicbnd4 26515 . . . . . . 7 (𝐴 ∈ ℝ+ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ≀ (1 / 𝐴))
1052, 104syl 17 . . . . . 6 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ≀ (1 / 𝐴))
1068nnrecred 12263 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (1 / π‘š) ∈ ℝ)
1071, 106fsumrecl 15680 . . . . . . . . . 10 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) ∈ ℝ)
108107, 40resubcld 11642 . . . . . . . . 9 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)) ∈ ℝ)
109108recnd 11242 . . . . . . . 8 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)) ∈ β„‚)
110109abscld 15383 . . . . . . 7 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ∈ ℝ)
1112rprecred 13027 . . . . . . 7 (πœ‘ β†’ (1 / 𝐴) ∈ ℝ)
112 0red 11217 . . . . . . . 8 (πœ‘ β†’ 0 ∈ ℝ)
113 1red 11215 . . . . . . . 8 (πœ‘ β†’ 1 ∈ ℝ)
114 0lt1 11736 . . . . . . . . 9 0 < 1
115114a1i 11 . . . . . . . 8 (πœ‘ β†’ 0 < 1)
116 loge 26095 . . . . . . . . 9 (logβ€˜e) = 1
117 mulog2sumlem1.3 . . . . . . . . . 10 (πœ‘ β†’ e ≀ 𝐴)
118 epr 16151 . . . . . . . . . . 11 e ∈ ℝ+
119 logleb 26111 . . . . . . . . . . 11 ((e ∈ ℝ+ ∧ 𝐴 ∈ ℝ+) β†’ (e ≀ 𝐴 ↔ (logβ€˜e) ≀ (logβ€˜π΄)))
120118, 2, 119sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ (e ≀ 𝐴 ↔ (logβ€˜e) ≀ (logβ€˜π΄)))
121117, 120mpbid 231 . . . . . . . . 9 (πœ‘ β†’ (logβ€˜e) ≀ (logβ€˜π΄))
122116, 121eqbrtrrid 5185 . . . . . . . 8 (πœ‘ β†’ 1 ≀ (logβ€˜π΄))
123112, 113, 11, 115, 122ltletrd 11374 . . . . . . 7 (πœ‘ β†’ 0 < (logβ€˜π΄))
124 lemul2 12067 . . . . . . 7 (((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ ∧ ((logβ€˜π΄) ∈ ℝ ∧ 0 < (logβ€˜π΄))) β†’ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ≀ (1 / 𝐴) ↔ ((logβ€˜π΄) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))) ≀ ((logβ€˜π΄) Β· (1 / 𝐴))))
125110, 111, 11, 123, 124syl112anc 1375 . . . . . 6 (πœ‘ β†’ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) ≀ (1 / 𝐴) ↔ ((logβ€˜π΄) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))) ≀ ((logβ€˜π΄) Β· (1 / 𝐴))))
126105, 125mpbid 231 . . . . 5 (πœ‘ β†’ ((logβ€˜π΄) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))) ≀ ((logβ€˜π΄) Β· (1 / 𝐴)))
12745rpcnd 13018 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ π‘š ∈ β„‚)
12845rpne0d 13021 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ π‘š β‰  0)
12963, 127, 128divrecd 11993 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ ((logβ€˜π΄) / π‘š) = ((logβ€˜π΄) Β· (1 / π‘š)))
130129sumeq2dv 15649 . . . . . . . . . 10 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) Β· (1 / π‘š)))
131106recnd 11242 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ (1...(βŒŠβ€˜π΄))) β†’ (1 / π‘š) ∈ β„‚)
1321, 38, 131fsummulc2 15730 . . . . . . . . . 10 (πœ‘ β†’ ((logβ€˜π΄) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š)) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) Β· (1 / π‘š)))
133130, 132eqtr4d 2776 . . . . . . . . 9 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) = ((logβ€˜π΄) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š)))
134133oveq1d 7424 . . . . . . . 8 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) = (((logβ€˜π΄) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š)) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))))
1351, 131fsumcl 15679 . . . . . . . . 9 (πœ‘ β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) ∈ β„‚)
13638, 135, 41subdid 11670 . . . . . . . 8 (πœ‘ β†’ ((logβ€˜π΄) Β· (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))) = (((logβ€˜π΄) Β· Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š)) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))))
137134, 136eqtr4d 2776 . . . . . . 7 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³))) = ((logβ€˜π΄) Β· (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³))))
138137fveq2d 6896 . . . . . 6 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) = (absβ€˜((logβ€˜π΄) Β· (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))))
139135, 41subcld 11571 . . . . . . 7 (πœ‘ β†’ (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)) ∈ β„‚)
14038, 139absmuld 15401 . . . . . 6 (πœ‘ β†’ (absβ€˜((logβ€˜π΄) Β· (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))) = ((absβ€˜(logβ€˜π΄)) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))))
141112, 11, 123ltled 11362 . . . . . . . 8 (πœ‘ β†’ 0 ≀ (logβ€˜π΄))
14211, 141absidd 15369 . . . . . . 7 (πœ‘ β†’ (absβ€˜(logβ€˜π΄)) = (logβ€˜π΄))
143142oveq1d 7424 . . . . . 6 (πœ‘ β†’ ((absβ€˜(logβ€˜π΄)) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))) = ((logβ€˜π΄) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))))
144138, 140, 1433eqtrd 2777 . . . . 5 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) = ((logβ€˜π΄) Β· (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))(1 / π‘š) βˆ’ ((logβ€˜π΄) + Ξ³)))))
1452rpcnd 13018 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ β„‚)
1462rpne0d 13021 . . . . . 6 (πœ‘ β†’ 𝐴 β‰  0)
14738, 145, 146divrecd 11993 . . . . 5 (πœ‘ β†’ ((logβ€˜π΄) / 𝐴) = ((logβ€˜π΄) Β· (1 / 𝐴)))
148126, 144, 1473brtr4d 5181 . . . 4 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) ≀ ((logβ€˜π΄) / 𝐴))
149 fveq2 6892 . . . . . . . . . . . . . 14 (𝑖 = π‘š β†’ (logβ€˜π‘–) = (logβ€˜π‘š))
150 id 22 . . . . . . . . . . . . . 14 (𝑖 = π‘š β†’ 𝑖 = π‘š)
151149, 150oveq12d 7427 . . . . . . . . . . . . 13 (𝑖 = π‘š β†’ ((logβ€˜π‘–) / 𝑖) = ((logβ€˜π‘š) / π‘š))
152151cbvsumv 15642 . . . . . . . . . . . 12 Σ𝑖 ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘–) / 𝑖) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘š) / π‘š)
153 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 β†’ (βŒŠβ€˜π‘¦) = (βŒŠβ€˜π΄))
154153oveq2d 7425 . . . . . . . . . . . . 13 (𝑦 = 𝐴 β†’ (1...(βŒŠβ€˜π‘¦)) = (1...(βŒŠβ€˜π΄)))
155154sumeq1d 15647 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ Ξ£π‘š ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘š) / π‘š) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š))
156152, 155eqtrid 2785 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ Σ𝑖 ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘–) / 𝑖) = Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š))
157 fveq2 6892 . . . . . . . . . . . . 13 (𝑦 = 𝐴 β†’ (logβ€˜π‘¦) = (logβ€˜π΄))
158157oveq1d 7424 . . . . . . . . . . . 12 (𝑦 = 𝐴 β†’ ((logβ€˜π‘¦)↑2) = ((logβ€˜π΄)↑2))
159158oveq1d 7424 . . . . . . . . . . 11 (𝑦 = 𝐴 β†’ (((logβ€˜π‘¦)↑2) / 2) = (((logβ€˜π΄)↑2) / 2))
160156, 159oveq12d 7427 . . . . . . . . . 10 (𝑦 = 𝐴 β†’ (Σ𝑖 ∈ (1...(βŒŠβ€˜π‘¦))((logβ€˜π‘–) / 𝑖) βˆ’ (((logβ€˜π‘¦)↑2) / 2)) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)))
161 ovex 7442 . . . . . . . . . 10 (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)) ∈ V
162160, 19, 161fvmpt 6999 . . . . . . . . 9 (𝐴 ∈ ℝ+ β†’ (πΉβ€˜π΄) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)))
1632, 162syl 17 . . . . . . . 8 (πœ‘ β†’ (πΉβ€˜π΄) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)))
164163oveq1d 7424 . . . . . . 7 (πœ‘ β†’ ((πΉβ€˜π΄) βˆ’ 𝐿) = ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)) βˆ’ 𝐿))
16549, 50, 51subsub4d 11602 . . . . . . 7 (πœ‘ β†’ ((Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ (((logβ€˜π΄)↑2) / 2)) βˆ’ 𝐿) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))
166164, 165eqtrd 2773 . . . . . 6 (πœ‘ β†’ ((πΉβ€˜π΄) βˆ’ 𝐿) = (Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))
167166fveq2d 6896 . . . . 5 (πœ‘ β†’ (absβ€˜((πΉβ€˜π΄) βˆ’ 𝐿)) = (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))))
16820simp3i 1142 . . . . . 6 ((𝐹 β‡π‘Ÿ 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ e ≀ 𝐴) β†’ (absβ€˜((πΉβ€˜π΄) βˆ’ 𝐿)) ≀ ((logβ€˜π΄) / 𝐴))
16924, 2, 117, 168syl3anc 1372 . . . . 5 (πœ‘ β†’ (absβ€˜((πΉβ€˜π΄) βˆ’ 𝐿)) ≀ ((logβ€˜π΄) / 𝐴))
170167, 169eqbrtrrd 5173 . . . 4 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿))) ≀ ((logβ€˜π΄) / 𝐴))
17144, 54, 57, 57, 148, 170le2addd 11833 . . 3 (πœ‘ β†’ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) + (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))) ≀ (((logβ€˜π΄) / 𝐴) + ((logβ€˜π΄) / 𝐴)))
17257recnd 11242 . . . 4 (πœ‘ β†’ ((logβ€˜π΄) / 𝐴) ∈ β„‚)
1731722timesd 12455 . . 3 (πœ‘ β†’ (2 Β· ((logβ€˜π΄) / 𝐴)) = (((logβ€˜π΄) / 𝐴) + ((logβ€˜π΄) / 𝐴)))
174171, 173breqtrrd 5177 . 2 (πœ‘ β†’ ((absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π΄) / π‘š) βˆ’ ((logβ€˜π΄) Β· ((logβ€˜π΄) + Ξ³)))) + (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜π‘š) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + 𝐿)))) ≀ (2 Β· ((logβ€˜π΄) / 𝐴)))
17533, 55, 59, 103, 174letrd 11371 1 (πœ‘ β†’ (absβ€˜(Ξ£π‘š ∈ (1...(βŒŠβ€˜π΄))((logβ€˜(𝐴 / π‘š)) / π‘š) βˆ’ ((((logβ€˜π΄)↑2) / 2) + ((Ξ³ Β· (logβ€˜π΄)) βˆ’ 𝐿)))) ≀ (2 Β· ((logβ€˜π΄) / 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  supcsup 9435  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   Β· cmul 11115  +∞cpnf 11245  β„*cxr 11247   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444   / cdiv 11871  β„•cn 12212  2c2 12267  β„+crp 12974  ...cfz 13484  βŒŠcfl 13755  β†‘cexp 14027  abscabs 15181   β‡π‘Ÿ crli 15429  Ξ£csu 15632  eceu 16006  logclog 26063  Ξ³cem 26496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-xnn0 12545  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ioc 13329  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-seq 13967  df-exp 14028  df-fac 14234  df-bc 14263  df-hash 14291  df-shft 15014  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-sum 15633  df-ef 16011  df-e 16012  df-sin 16013  df-cos 16014  df-tan 16015  df-pi 16016  df-dvds 16198  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lp 22640  df-perf 22641  df-cn 22731  df-cnp 22732  df-haus 22819  df-cmp 22891  df-tx 23066  df-hmeo 23259  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-xms 23826  df-ms 23827  df-tms 23828  df-cncf 24394  df-limc 25383  df-dv 25384  df-ulm 25889  df-log 26065  df-cxp 26066  df-atan 26372  df-em 26497
This theorem is referenced by:  mulog2sumlem2  27038
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