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Theorem logfac2 25233
Description: Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Assertion
Ref Expression
logfac2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
Distinct variable group:   𝐴,𝑘

Proof of Theorem logfac2
Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flge0nn0 12829 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
2 logfac 24638 . . 3 ((⌊‘𝐴) ∈ ℕ0 → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
31, 2syl 17 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
4 fzfid 12980 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
5 fzfid 12980 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ∈ Fin)
6 ssrab2 3847 . . . . 5 {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ⊆ (1...(⌊‘𝐴))
7 ssfi 8387 . . . . 5 (((1...(⌊‘𝐴)) ∈ Fin ∧ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ⊆ (1...(⌊‘𝐴))) → {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ∈ Fin)
85, 6, 7sylancl 580 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ∈ Fin)
9 flcl 12804 . . . . . . . . 9 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
109adantr 472 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ)
11 fznn 12615 . . . . . . . 8 ((⌊‘𝐴) ∈ ℤ → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴))))
1210, 11syl 17 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴))))
1312anbi1d 623 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
14 nnre 11282 . . . . . . . . . . 11 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
1514ad2antlr 718 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℝ)
16 elfznn 12577 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
1716ad2antrl 719 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℕ)
1817nnred 11291 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℝ)
19 reflcl 12805 . . . . . . . . . . 11 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
2019ad3antrrr 721 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → (⌊‘𝐴) ∈ ℝ)
21 simprr 789 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘𝑛)
22 nnz 11646 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2322ad2antlr 718 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℤ)
24 dvdsle 15319 . . . . . . . . . . . 12 ((𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑘𝑛𝑘𝑛))
2523, 17, 24syl2anc 579 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → (𝑘𝑛𝑘𝑛))
2621, 25mpd 15 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘𝑛)
27 elfzle2 12552 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ≤ (⌊‘𝐴))
2827ad2antrl 719 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ≤ (⌊‘𝐴))
2915, 18, 20, 26, 28letrd 10448 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴))
3029expl 449 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴)))
3130pm4.71rd 558 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))))
32 an12 635 . . . . . . 7 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
33 anass 460 . . . . . . . 8 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
34 an12 635 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3533, 34bitri 266 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3631, 32, 353bitr4g 305 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3713, 36bitr4d 273 . . . . 5 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛))))
38 breq2 4813 . . . . . . 7 (𝑥 = 𝑛 → (𝑘𝑥𝑘𝑛))
3938elrab 3519 . . . . . 6 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))
4039anbi2i 616 . . . . 5 ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
41 breq1 4812 . . . . . . 7 (𝑥 = 𝑘 → (𝑥𝑛𝑘𝑛))
4241elrab 3519 . . . . . 6 (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑘 ∈ ℕ ∧ 𝑘𝑛))
4342anbi2i 616 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)))
4437, 40, 433bitr4g 305 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})))
45 elfznn 12577 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
4645adantl 473 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
47 vmacl 25135 . . . . . . 7 (𝑘 ∈ ℕ → (Λ‘𝑘) ∈ ℝ)
4846, 47syl 17 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℝ)
4948recnd 10322 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℂ)
5049adantrr 708 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})) → (Λ‘𝑘) ∈ ℂ)
514, 4, 8, 44, 50fsumcom2 14792 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘))
52 fsumconst 14808 . . . . . 6 (({𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ∈ Fin ∧ (Λ‘𝑘) ∈ ℂ) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)))
538, 49, 52syl2anc 579 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)))
54 fzfid 12980 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ∈ Fin)
55 simpll 783 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
56 eqid 2765 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚)) = (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚))
5755, 46, 56dvdsflf1o 25204 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚)):(1...(⌊‘(𝐴 / 𝑘)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})
58 f1oeng 8179 . . . . . . . . 9 (((1...(⌊‘(𝐴 / 𝑘))) ∈ Fin ∧ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚)):(1...(⌊‘(𝐴 / 𝑘)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) → (1...(⌊‘(𝐴 / 𝑘))) ≈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})
5954, 57, 58syl2anc 579 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ≈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})
60 hasheni 13340 . . . . . . . 8 ((1...(⌊‘(𝐴 / 𝑘))) ≈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} → (♯‘(1...(⌊‘(𝐴 / 𝑘)))) = (♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}))
6159, 60syl 17 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (♯‘(1...(⌊‘(𝐴 / 𝑘)))) = (♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}))
62 simpl 474 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
63 nndivre 11313 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℝ)
6462, 45, 63syl2an 589 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑘) ∈ ℝ)
65 nngt0 11306 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 0 < 𝑘)
6614, 65jca 507 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
6745, 66syl 17 . . . . . . . . . 10 (𝑘 ∈ (1...(⌊‘𝐴)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
68 divge0 11146 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝐴 / 𝑘))
6967, 68sylan2 586 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ (𝐴 / 𝑘))
70 flge0nn0 12829 . . . . . . . . 9 (((𝐴 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑘)) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
7164, 69, 70syl2anc 579 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
72 hashfz1 13338 . . . . . . . 8 ((⌊‘(𝐴 / 𝑘)) ∈ ℕ0 → (♯‘(1...(⌊‘(𝐴 / 𝑘)))) = (⌊‘(𝐴 / 𝑘)))
7371, 72syl 17 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (♯‘(1...(⌊‘(𝐴 / 𝑘)))) = (⌊‘(𝐴 / 𝑘)))
7461, 73eqtr3d 2801 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) = (⌊‘(𝐴 / 𝑘)))
7574oveq1d 6857 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((♯‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)) = ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)))
7664flcld 12807 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℤ)
7776zcnd 11730 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℂ)
7877, 49mulcomd 10315 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
7953, 75, 783eqtrd 2803 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8079sumeq2dv 14720 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8116adantl 473 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
82 vmasum 25232 . . . . 5 (𝑛 ∈ ℕ → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘) = (log‘𝑛))
8381, 82syl 17 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘) = (log‘𝑛))
8483sumeq2dv 14720 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
8551, 80, 843eqtr3d 2807 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
863, 85eqtr4d 2802 1 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {crab 3059  wss 3732   class class class wbr 4809  cmpt 4888  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  cen 8157  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   · cmul 10194   < clt 10328  cle 10329   / cdiv 10938  cn 11274  0cn0 11538  cz 11624  ...cfz 12533  cfl 12799  !cfa 13264  chash 13321  Σcsu 14703  cdvds 15267  logclog 24592  Λcvma 25109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-ioo 12381  df-ioc 12382  df-ico 12383  df-icc 12384  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-shft 14094  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-limsup 14489  df-clim 14506  df-rlim 14507  df-sum 14704  df-ef 15082  df-sin 15084  df-cos 15085  df-pi 15087  df-dvds 15268  df-gcd 15500  df-prm 15668  df-pc 15823  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-starv 16231  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-unif 16239  df-hom 16240  df-cco 16241  df-rest 16351  df-topn 16352  df-0g 16370  df-gsum 16371  df-topgen 16372  df-pt 16373  df-prds 16376  df-xrs 16430  df-qtop 16435  df-imas 16436  df-xps 16438  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-submnd 17604  df-mulg 17810  df-cntz 18015  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-fbas 20016  df-fg 20017  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-perf 21221  df-cn 21311  df-cnp 21312  df-haus 21399  df-tx 21645  df-hmeo 21838  df-fil 21929  df-fm 22021  df-flim 22022  df-flf 22023  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-limc 23921  df-dv 23922  df-log 24594  df-vma 25115
This theorem is referenced by:  vmadivsum  25462
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