Theorem dchrvmasumlem1 27617

Description: An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)

Hypotheses

Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasum.a	⊢ (𝜑 → 𝐴 ∈ ℝ+)

Assertion

Ref	Expression
dchrvmasumlem1	⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))

Distinct variable groups:   𝑚,𝑛, 1   𝑚,𝑑,𝑛,𝐴   𝑚,𝑁,𝑛   𝜑,𝑑,𝑚,𝑛   𝑚,𝑍,𝑛   𝐷,𝑚,𝑛   𝐿,𝑑,𝑚,𝑛   𝑋,𝑑,𝑚,𝑛   𝐴,𝑛

Allowed substitution hints:   𝐷(𝑑)   1 (𝑑)   𝐺(𝑚,𝑛,𝑑)   𝑁(𝑑)   𝑍(𝑑)

Proof of Theorem dchrvmasumlem1

Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6876	. . . 4 ⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚))))
2		oveq2 7408	. . . . 5 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) / 𝑛) = ((μ‘𝑑) / (𝑑 · 𝑚)))
3		fvoveq1 7423	. . . . 5 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
4	2, 3	oveq12d 7418	. . . 4 ⊢ (𝑛 = (𝑑 · 𝑚) → (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))) = (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑))))
5	1, 4	oveq12d 7418	. . 3 ⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
6		dchrvmasum.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+)
7	6	rpred 13051	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		rpvmasum.g	. . . . . 6 ⊢ 𝐺 = (DChr‘𝑁)
9		rpvmasum.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		rpvmasum.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
11		rpvmasum.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
12		dchrisum.b	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
14		elfzelz 13543	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℤ)
15	14	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℤ)
16	8, 9, 10, 11, 13, 15	dchrzrhcl 27367	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
17	16	adantrr 729	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
18		elrabi 3649	. . . . . . . . . 10 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} → 𝑑 ∈ ℕ)
19	18	ad2antll 741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝑑 ∈ ℕ)
20		mucl 27263	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
21	19, 20	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ)
22	21	zred 12691	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (μ‘𝑑) ∈ ℝ)
23		elfznn 13572	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
24	23	ad2antrl 740	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝑛 ∈ ℕ)
25	22, 24	nndivred 12281	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → ((μ‘𝑑) / 𝑛) ∈ ℝ)
26	25	recnd 11225	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → ((μ‘𝑑) / 𝑛) ∈ ℂ)
27	24	nnrpd 13049	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝑛 ∈ ℝ+)
28	19	nnrpd 13049	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝑑 ∈ ℝ+)
29	27, 28	rpdivcld 13068	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+)
30	29	relogcld 26746	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (log‘(𝑛 / 𝑑)) ∈ ℝ)
31	30	recnd 11225	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (log‘(𝑛 / 𝑑)) ∈ ℂ)
32	26, 31	mulcld 11217	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))) ∈ ℂ)
33	17, 32	mulcld 11217	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))) ∈ ℂ)
34	5, 7, 33	dvdsflsumcom 27310	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
35		vmaf 27241	. . . . . . . . . . . . 13 ⊢ Λ:ℕ⟶ℝ
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → Λ:ℕ⟶ℝ)
37		ax-resscn 11145	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
38		fss 6712	. . . . . . . . . . . 12 ⊢ ((Λ:ℕ⟶ℝ ∧ ℝ ⊆ ℂ) → Λ:ℕ⟶ℂ)
39	36, 37, 38	sylancl 597	. . . . . . . . . . 11 ⊢ (𝜑 → Λ:ℕ⟶ℂ)
40		vmasum 27338	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} (Λ‘𝑖) = (log‘𝑚))
41	40	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} (Λ‘𝑖) = (log‘𝑚))
42	41	eqcomd 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (log‘𝑚) = Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} (Λ‘𝑖))
43	42	mpteq2dva 5198	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (log‘𝑚)) = (𝑚 ∈ ℕ ↦ Σ𝑖 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} (Λ‘𝑖)))
44	39, 43	muinv 27315	. . . . . . . . . 10 ⊢ (𝜑 → Λ = (𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑)))))
45	44	fveq1d 6873	. . . . . . . . 9 ⊢ (𝜑 → (Λ‘𝑛) = ((𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))‘𝑛))
46		sumex 15729	. . . . . . . . . 10 ⊢ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))) ∈ V
47		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑)))) = (𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))
48	47	fvmpt2 6991	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))) ∈ V) → ((𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))‘𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))
49	23, 46, 48	sylancl 597	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → ((𝑛 ∈ ℕ ↦ Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))‘𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))
50	45, 49	sylan9eq 2820	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))))
51		breq1 5108	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑑 → (𝑥 ∥ 𝑛 ↔ 𝑑 ∥ 𝑛))
52	51	elrab 3653	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))
53	52	simprbi 502	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} → 𝑑 ∥ 𝑛)
54	53	adantl 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → 𝑑 ∥ 𝑛)
55	23	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
56		nndivdvds 16309	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ 𝑑 ∈ ℕ) → (𝑑 ∥ 𝑛 ↔ (𝑛 / 𝑑) ∈ ℕ))
57	55, 18, 56	syl2an 607	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (𝑑 ∥ 𝑛 ↔ (𝑛 / 𝑑) ∈ ℕ))
58	54, 57	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
59		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 / 𝑑) → (log‘𝑚) = (log‘(𝑛 / 𝑑)))
60		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ ↦ (log‘𝑚)) = (𝑚 ∈ ℕ ↦ (log‘𝑚))
61		fvex 6884	. . . . . . . . . . . 12 ⊢ (log‘(𝑛 / 𝑑)) ∈ V
62	59, 60, 61	fvmpt 6979	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℕ → ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑)) = (log‘(𝑛 / 𝑑)))
63	58, 62	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑)) = (log‘(𝑛 / 𝑑)))
64	63	oveq2d 7416	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))) = ((μ‘𝑑) · (log‘(𝑛 / 𝑑))))
65	64	sumeq2dv 15743	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · ((𝑚 ∈ ℕ ↦ (log‘𝑚))‘(𝑛 / 𝑑))) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (log‘(𝑛 / 𝑑))))
66	50, 65	eqtrd 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Λ‘𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (log‘(𝑛 / 𝑑))))
67	66	oveq1d 7415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) / 𝑛) = (Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (log‘(𝑛 / 𝑑))) / 𝑛))
68		fzfid 14000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin)
69		dvdsssfz1 16366	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛))
70	55, 69	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛))
71	68, 70	ssfid 9217	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ∈ Fin)
72	55	nncnd 12240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
73	21	zcnd 12692	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
74	73	anassrs 472	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (μ‘𝑑) ∈ ℂ)
75	31	anassrs 472	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (log‘(𝑛 / 𝑑)) ∈ ℂ)
76	74, 75	mulcld 11217	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → ((μ‘𝑑) · (log‘(𝑛 / 𝑑))) ∈ ℂ)
77	55	nnne0d 12277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
78	71, 72, 76, 77	fsumdivc 15827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((μ‘𝑑) · (log‘(𝑛 / 𝑑))) / 𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) · (log‘(𝑛 / 𝑑))) / 𝑛))
79	18	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → 𝑑 ∈ ℕ)
80	79, 20	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (μ‘𝑑) ∈ ℤ)
81	80	zcnd 12692	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (μ‘𝑑) ∈ ℂ)
82	72	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → 𝑛 ∈ ℂ)
83	77	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → 𝑛 ≠ 0)
84	81, 75, 82, 83	div23d 12019	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (((μ‘𝑑) · (log‘(𝑛 / 𝑑))) / 𝑛) = (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))))
85	84	sumeq2dv 15743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) · (log‘(𝑛 / 𝑑))) / 𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))))
86	67, 78, 85	3eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((Λ‘𝑛) / 𝑛) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))))
87	86	oveq2d 7416	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = ((𝑋‘(𝐿‘𝑛)) · Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))))
88	32	anassrs 472	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) → (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑))) ∈ ℂ)
89	71, 16, 88	fsummulc2 15825	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) · Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))))
90	87, 89	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))))
91	90	sumeq2dv 15743	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ((𝑋‘(𝐿‘𝑛)) · (((μ‘𝑑) / 𝑛) · (log‘(𝑛 / 𝑑)))))
92		fzfid 14000	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin)
93	12	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑋 ∈ 𝐷)
94		elfzelz 13543	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝐴)) → 𝑑 ∈ ℤ)
95	94	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℤ)
96	8, 9, 10, 11, 93, 95	dchrzrhcl 27367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
97		fznnfl 13886	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴)))
98	7, 97	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴)))
99	98	simprbda 503	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ)
100	99, 20	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (μ‘𝑑) ∈ ℤ)
101	100	zred 12691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (μ‘𝑑) ∈ ℝ)
102	101, 99	nndivred 12281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
103	102	recnd 11225	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
104	96, 103	mulcld 11217	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
105	12	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑋 ∈ 𝐷)
106		elfzelz 13543	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℤ)
107	106	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℤ)
108	8, 9, 10, 11, 105, 107	dchrzrhcl 27367	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
109		elfznn 13572	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑))) → 𝑚 ∈ ℕ)
110	109	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℕ)
111	110	nnrpd 13049	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℝ+)
112	111	relogcld 26746	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℝ)
113	112, 110	nndivred 12281	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℝ)
114	113	recnd 11225	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((log‘𝑚) / 𝑚) ∈ ℂ)
115	108, 114	mulcld 11217	. . . . 5 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚)) ∈ ℂ)
116	92, 104, 115	fsummulc2 15825	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))
117	96	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
118	103	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
119	117, 118, 108, 114	mul4d 11410	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) · (((μ‘𝑑) / 𝑑) · ((log‘𝑚) / 𝑚))))
120	94	ad2antlr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℤ)
121	8, 9, 10, 11, 105, 120, 107	dchrzrhmul 27368	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑋‘(𝐿‘(𝑑 · 𝑚))) = ((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))))
122	101	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (μ‘𝑑) ∈ ℝ)
123	122	recnd 11225	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (μ‘𝑑) ∈ ℂ)
124	112	recnd 11225	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘𝑚) ∈ ℂ)
125	99	nnrpd 13049	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℝ+)
126	125	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℝ+)
127	126, 111	rpmulcld 13067	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑑 · 𝑚) ∈ ℝ+)
128	127	rpcnne0d 13060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑑 · 𝑚) ∈ ℂ ∧ (𝑑 · 𝑚) ≠ 0))
129		div23 11879	. . . . . . . . 9 ⊢ (((μ‘𝑑) ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ ((𝑑 · 𝑚) ∈ ℂ ∧ (𝑑 · 𝑚) ≠ 0)) → (((μ‘𝑑) · (log‘𝑚)) / (𝑑 · 𝑚)) = (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘𝑚)))
130	123, 124, 128, 129	syl3anc 1394	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) · (log‘𝑚)) / (𝑑 · 𝑚)) = (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘𝑚)))
131	126	rpcnne0d 13060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
132	111	rpcnne0d 13060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
133		divmuldiv 11906	. . . . . . . . 9 ⊢ ((((μ‘𝑑) ∈ ℂ ∧ (log‘𝑚) ∈ ℂ) ∧ ((𝑑 ∈ ℂ ∧ 𝑑 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))) → (((μ‘𝑑) / 𝑑) · ((log‘𝑚) / 𝑚)) = (((μ‘𝑑) · (log‘𝑚)) / (𝑑 · 𝑚)))
134	123, 124, 131, 132, 133	syl22anc 851	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) / 𝑑) · ((log‘𝑚) / 𝑚)) = (((μ‘𝑑) · (log‘𝑚)) / (𝑑 · 𝑚)))
135	110	nncnd 12240	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑚 ∈ ℂ)
136	126	rpcnd 13053	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ∈ ℂ)
137	126	rpne0d 13056	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → 𝑑 ≠ 0)
138	135, 136, 137	divcan3d 11987	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
139	138	fveq2d 6875	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
140	139	oveq2d 7416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑))) = (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘𝑚)))
141	130, 134, 140	3eqtr4rd 2811	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑))) = (((μ‘𝑑) / 𝑑) · ((log‘𝑚) / 𝑚)))
142	121, 141	oveq12d 7418	. . . . . 6 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) · (((μ‘𝑑) / 𝑑) · ((log‘𝑚) / 𝑚))))
143	119, 142	eqtr4d 2803	. . . . 5 ⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
144	143	sumeq2dv 15743	. . . 4 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
145	116, 144	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
146	145	sumeq2dv 15743	. 2 ⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))) = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘(𝑑 · 𝑚))) · (((μ‘𝑑) / (𝑑 · 𝑚)) · (log‘((𝑑 · 𝑚) / 𝑑)))))
147	34, 91, 146	3eqtr4d 2810	1 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝐴))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))((𝑋‘(𝐿‘𝑚)) · ((log‘𝑚) / 𝑚))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {crab 3417  Vcvv 3457   ⊆ wss 3907   class class class wbr 5105   ↦ cmpt 5186  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  ℝcr 11087  0cc0 11088  1c1 11089   · cmul 11093   ≤ cle 11232   / cdiv 11859  ℕcn 12224  ℤcz 12582  ℝ⁺crp 13007  ...cfz 13526  ⌊cfl 13814  Σcsu 15727   ∥ cdvds 16300  Basecbs 17259  0_gc0g 17482  ℤRHomczrh 21609  ℤ/nℤczn 21612  logclog 26677  Λcvma 27214  μcmu 27217  DChrcdchr 27354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166  ax-addf 11167  ax-mulf 11168

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-disj 5073  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-tpos 8210  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-er 8682  df-ec 8684  df-qs 8688  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-fi 9359  df-sup 9390  df-inf 9391  df-oi 9460  df-dju 9875  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-xnn0 12569  df-z 12583  df-dec 12703  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13128  df-xadd 13129  df-xmul 13130  df-ioo 13367  df-ioc 13368  df-ico 13369  df-icc 13370  df-fz 13527  df-fzo 13674  df-fl 13816  df-mod 13894  df-seq 14029  df-exp 14089  df-fac 14301  df-bc 14330  df-hash 14358  df-shft 15094  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-limsup 15512  df-clim 15529  df-rlim 15530  df-sum 15728  df-ef 16111  df-sin 16113  df-cos 16114  df-pi 16116  df-dvds 16301  df-gcd 16543  df-prm 16720  df-pc 16887  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-starv 17315  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-unif 17323  df-hom 17324  df-cco 17325  df-rest 17465  df-topn 17466  df-0g 17484  df-gsum 17485  df-topgen 17486  df-pt 17487  df-prds 17490  df-xrs 17546  df-qtop 17551  df-imas 17552  df-qus 17553  df-xps 17554  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-mulg 19125  df-subg 19180  df-nsg 19181  df-eqg 19182  df-ghm 19275  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-oppr 20410  df-dvdsr 20430  df-unit 20431  df-rhm 20545  df-subrng 20622  df-subrg 20646  df-lmod 20952  df-lss 21022  df-lsp 21062  df-sra 21263  df-rgmod 21264  df-lidl 21301  df-rsp 21302  df-2idl 21351  df-psmet 21474  df-xmet 21475  df-met 21476  df-bl 21477  df-mopn 21478  df-fbas 21479  df-fg 21480  df-cnfld 21483  df-zring 21557  df-zrh 21613  df-zn 21616  df-top 23012  df-topon 23029  df-topsp 23051  df-bases 23064  df-cld 23137  df-ntr 23138  df-cls 23139  df-nei 23216  df-lp 23254  df-perf 23255  df-cn 23345  df-cnp 23346  df-haus 23433  df-tx 23680  df-hmeo 23873  df-fil 23964  df-fm 24056  df-flim 24057  df-flf 24058  df-xms 24438  df-ms 24439  df-tms 24440  df-cncf 24998  df-limc 25986  df-dv 25987  df-log 26679  df-vma 27220  df-mu 27223  df-dchr 27355

This theorem is referenced by:  dchrvmasum2if  27619