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Theorem logfac2 24876
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 12577 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ0)
2 logfac 24285 . . 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 12728 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (1...(⌊‘𝐴)) ∈ Fin)
5 fzfid 12728 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘𝐴)) ∈ Fin)
6 ssrab2 3672 . . . . 5 {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ⊆ (1...(⌊‘𝐴))
7 ssfi 8140 . . . . 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 12552 . . . . . . . . 9 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℤ)
109adantr 481 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℤ)
11 fznn 12366 . . . . . . . 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 740 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
14 nnre 10987 . . . . . . . . . . 11 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
1514ad2antlr 762 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℝ)
16 elfznn 12328 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
1716ad2antrl 763 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℕ)
1817nnred 10995 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ∈ ℝ)
19 reflcl 12553 . . . . . . . . . . 11 (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
2019ad3antrrr 765 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → (⌊‘𝐴) ∈ ℝ)
21 simprr 795 . . . . . . . . . . 11 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘𝑛)
22 nnz 11359 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
2322ad2antlr 762 . . . . . . . . . . . 12 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ∈ ℤ)
24 dvdsle 14975 . . . . . . . . . . . 12 ((𝑘 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑘𝑛𝑘𝑛))
2523, 17, 24syl2anc 692 . . . . . . . . . . 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 12303 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ≤ (⌊‘𝐴))
2827ad2antrl 763 . . . . . . . . . 10 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑛 ≤ (⌊‘𝐴))
2915, 18, 20, 26, 28letrd 10154 . . . . . . . . 9 ((((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴))
3029expl 647 . . . . . . . 8 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) → 𝑘 ≤ (⌊‘𝐴)))
3130pm4.71rd 666 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))))
32 an12 837 . . . . . . 7 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
33 anass 680 . . . . . . . 8 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
34 an12 837 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3533, 34bitri 264 . . . . . . 7 (((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑘 ≤ (⌊‘𝐴) ∧ (𝑘 ∈ ℕ ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3631, 32, 353bitr4g 303 . . . . . 6 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)) ↔ ((𝑘 ∈ ℕ ∧ 𝑘 ≤ (⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))))
3713, 36bitr4d 271 . . . . 5 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛))))
38 breq2 4627 . . . . . . 7 (𝑥 = 𝑛 → (𝑘𝑥𝑘𝑛))
3938elrab 3351 . . . . . 6 (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛))
4039anbi2i 729 . . . . 5 ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘𝑛)))
41 breq1 4626 . . . . . . 7 (𝑥 = 𝑘 → (𝑥𝑛𝑘𝑛))
4241elrab 3351 . . . . . 6 (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} ↔ (𝑘 ∈ ℕ ∧ 𝑘𝑛))
4342anbi2i 729 . . . . 5 ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑘 ∈ ℕ ∧ 𝑘𝑛)))
4437, 40, 433bitr4g 303 . . . 4 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛})))
45 elfznn 12328 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝐴)) → 𝑘 ∈ ℕ)
4645adantl 482 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝑘 ∈ ℕ)
47 vmacl 24778 . . . . . . 7 (𝑘 ∈ ℕ → (Λ‘𝑘) ∈ ℝ)
4846, 47syl 17 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℝ)
4948recnd 10028 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑘) ∈ ℂ)
5049adantrr 752 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})) → (Λ‘𝑘) ∈ ℂ)
514, 4, 8, 44, 50fsumcom2 14452 . . 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 14469 . . . . . 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 692 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)))
54 fzfid 12728 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ∈ Fin)
55 simpll 789 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
56 eqid 2621 . . . . . . . . . 10 (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚)) = (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑘))) ↦ (𝑘 · 𝑚))
5755, 46, 56dvdsflf1o 24847 . . . . . . . . 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 7934 . . . . . . . . 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 692 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑘))) ≈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥})
60 hasheni 13092 . . . . . . . 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 473 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → 𝐴 ∈ ℝ)
63 nndivre 11016 . . . . . . . . . 10 ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ) → (𝐴 / 𝑘) ∈ ℝ)
6462, 45, 63syl2an 494 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (𝐴 / 𝑘) ∈ ℝ)
65 nngt0 11009 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 0 < 𝑘)
6614, 65jca 554 . . . . . . . . . . 11 (𝑘 ∈ ℕ → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
6745, 66syl 17 . . . . . . . . . 10 (𝑘 ∈ (1...(⌊‘𝐴)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
68 divge0 10852 . . . . . . . . . 10 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ (𝐴 / 𝑘))
6967, 68sylan2 491 . . . . . . . . 9 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → 0 ≤ (𝐴 / 𝑘))
70 flge0nn0 12577 . . . . . . . . 9 (((𝐴 / 𝑘) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑘)) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
7164, 69, 70syl2anc 692 . . . . . . . 8 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℕ0)
72 hashfz1 13090 . . . . . . . 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 2657 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) = (⌊‘(𝐴 / 𝑘)))
7574oveq1d 6630 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((#‘{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥}) · (Λ‘𝑘)) = ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)))
7664flcld 12555 . . . . . . 7 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℤ)
7776zcnd 11443 . . . . . 6 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → (⌊‘(𝐴 / 𝑘)) ∈ ℂ)
7877, 49mulcomd 10021 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → ((⌊‘(𝐴 / 𝑘)) · (Λ‘𝑘)) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
7953, 75, 783eqtrd 2659 . . . 4 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑘 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = ((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8079sumeq2dv 14383 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑘𝑥} (Λ‘𝑘) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))))
8116adantl 482 . . . . 5 (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
82 vmasum 24875 . . . . 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 14383 . . 3 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥𝑛} (Λ‘𝑘) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
8551, 80, 843eqtr3d 2663 . 2 ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘))) = Σ𝑛 ∈ (1...(⌊‘𝐴))(log‘𝑛))
863, 85eqtr4d 2658 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 196  wa 384   = wceq 1480  wcel 1987  {crab 2912  wss 3560   class class class wbr 4623  cmpt 4683  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cen 7912  Fincfn 7915  cc 9894  cr 9895  0cc0 9896  1c1 9897   · cmul 9901   < clt 10034  cle 10035   / cdiv 10644  cn 10980  0cn0 11252  cz 11337  ...cfz 12284  cfl 12547  !cfa 13016  #chash 13073  Σcsu 14366  cdvds 14926  logclog 24239  Λcvma 24752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975  ax-mulf 9976
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ioc 12138  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-fac 13017  df-bc 13046  df-hash 13074  df-shft 13757  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-limsup 14152  df-clim 14169  df-rlim 14170  df-sum 14367  df-ef 14742  df-sin 14744  df-cos 14745  df-pi 14747  df-dvds 14927  df-gcd 15160  df-prm 15329  df-pc 15485  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-starv 15896  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-unif 15905  df-hom 15906  df-cco 15907  df-rest 16023  df-topn 16024  df-0g 16042  df-gsum 16043  df-topgen 16044  df-pt 16045  df-prds 16048  df-xrs 16102  df-qtop 16107  df-imas 16108  df-xps 16110  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-mulg 17481  df-cntz 17690  df-cmn 18135  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-fbas 19683  df-fg 19684  df-cnfld 19687  df-top 20639  df-topon 20656  df-topsp 20677  df-bases 20690  df-cld 20763  df-ntr 20764  df-cls 20765  df-nei 20842  df-lp 20880  df-perf 20881  df-cn 20971  df-cnp 20972  df-haus 21059  df-tx 21305  df-hmeo 21498  df-fil 21590  df-fm 21682  df-flim 21683  df-flf 21684  df-xms 22065  df-ms 22066  df-tms 22067  df-cncf 22621  df-limc 23570  df-dv 23571  df-log 24241  df-vma 24758
This theorem is referenced by:  vmadivsum  25105
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