Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  logfac2 Structured version   Visualization version   GIF version

Theorem logfac2 24742
 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 12483 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
2 logfac 24151 . . 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 12634 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
5 fzfid 12634 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ∈ Fin)
6 ssrab2 3650 . . . . 5 {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ⊆ (1...(⌊‘𝐴))
7 ssfi 8065 . . . . 5 (((1...(⌊‘𝐴)) ∈ Fin ∧ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ⊆ (1...(⌊‘𝐴))) → {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ∈ Fin)
85, 6, 7sylancl 693 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ∈ Fin)
9 flcl 12458 . . . . . . . . 9 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
109adantr 480 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ)
11 fznn 12278 . . . . . . . 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 737 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
14 nnre 10904 . . . . . . . . . . 11 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
1514ad2antlr 759 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℝ)
16 elfznn 12241 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
1716ad2antrl 760 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℕ)
1817nnred 10912 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℝ)
19 reflcl 12459 . . . . . . . . . . 11 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
2019ad3antrrr 762 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → (⌊‘𝐴) ∈ ℝ)
21 simprr 792 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘𝑛)
22 nnz 11276 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2322ad2antlr 759 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℤ)
24 dvdsle 14870 . . . . . . . . . . . 12 ((𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑘𝑛𝑘𝑛))
2523, 17, 24syl2anc 691 . . . . . . . . . . 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 12216 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ≤ (⌊‘𝐴))
2827ad2antrl 760 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ≤ (⌊‘𝐴))
2915, 18, 20, 26, 28letrd 10073 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴))
3029expl 646 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴)))
3130pm4.71rd 665 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))))
32 an12 834 . . . . . . 7 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
33 anass 679 . . . . . . . 8 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
34 an12 834 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3533, 34bitri 263 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3631, 32, 353bitr4g 302 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3713, 36bitr4d 270 . . . . 5 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛))))
38 breq2 4587 . . . . . . 7 (𝑥 = 𝑛 → (𝑘𝑥𝑘𝑛))
3938elrab 3331 . . . . . 6 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))
4039anbi2i 726 . . . . 5 ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
41 breq1 4586 . . . . . . 7 (𝑥 = 𝑘 → (𝑥𝑛𝑘𝑛))
4241elrab 3331 . . . . . 6 (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑘 ∈ ℕ ∧ 𝑘𝑛))
4342anbi2i 726 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)))
4437, 40, 433bitr4g 302 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})))
45 elfznn 12241 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
4645adantl 481 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
47 vmacl 24644 . . . . . . 7 (𝑘 ∈ ℕ → (Λ‘𝑘) ∈ ℝ)
4846, 47syl 17 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℝ)
4948recnd 9947 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℂ)
5049adantrr 749 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})) → (Λ‘𝑘) ∈ ℂ)
514, 4, 8, 44, 50fsumcom2 14347 . . 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 14364 . . . . . 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 691 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)))
54 fzfid 12634 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ∈ Fin)
55 simpll 786 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
56 eqid 2610 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚)) = (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚))
5755, 46, 56dvdsflf1o 24713 . . . . . . . . 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 7860 . . . . . . . . 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 691 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ≈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})
60 hasheni 12998 . . . . . . . 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 472 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
63 nndivre 10933 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℝ)
6462, 45, 63syl2an 493 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑘) ∈ ℝ)
65 nngt0 10926 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 0 < 𝑘)
6614, 65jca 553 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
6745, 66syl 17 . . . . . . . . . 10 (𝑘 ∈ (1...(⌊‘𝐴)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
68 divge0 10771 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝐴 / 𝑘))
6967, 68sylan2 490 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ (𝐴 / 𝑘))
70 flge0nn0 12483 . . . . . . . . 9 (((𝐴 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑘)) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
7164, 69, 70syl2anc 691 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
72 hashfz1 12996 . . . . . . . 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 2646 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) = (⌊‘(𝐴 / 𝑘)))
7574oveq1d 6564 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)) = ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)))
7664flcld 12461 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℤ)
7776zcnd 11359 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℂ)
7877, 49mulcomd 9940 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
7953, 75, 783eqtrd 2648 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8079sumeq2dv 14281 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8116adantl 481 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
82 vmasum 24741 . . . . 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 14281 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
8551, 80, 843eqtr3d 2652 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
863, 85eqtr4d 2647 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  ⌊cfl 12453  !cfa 12922  #chash 12979  Σcsu 14264   ∥ cdvds 14821  logclog 24105  Λcvma 24618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-sin 14639  df-cos 14640  df-pi 14642  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-vma 24624 This theorem is referenced by:  vmadivsum  24971
 Copyright terms: Public domain W3C validator