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Theorem dchrmusum2 27626

Description: The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded, provided that 𝑇 ≠ 0. Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-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 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrisumn0.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrisumn0.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrisumn0.t	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
dchrisumn0.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))

**Assertion**
Ref	Expression
dchrmusum2	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑦, 1 𝑥,𝑑,𝑦,𝐶 𝐹,𝑑,𝑥,𝑦 𝑎,𝑑,𝑥,𝑦 𝑥,𝑁,𝑦 𝜑,𝑑,𝑥 𝑇,𝑑,𝑥,𝑦 𝑥,𝑍,𝑦 𝑥,𝐷,𝑦 𝐿,𝑎,𝑑,𝑥,𝑦 𝑋,𝑎,𝑑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎,𝑑) 𝑇(𝑎) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑍(𝑎,𝑑)

Proof of Theorem dchrmusum2 Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpssre 13026	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2		ax-1cn 11160	. . . 4 ⊢ 1 ∈ ℂ
3		o1const 15673	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
4	1, 2, 3	mp2an 704	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1)
5	4	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
6	2	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		fzfid 14011	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
8		rpvmasum.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
9		rpvmasum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
10		rpvmasum.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
11		rpvmasum.l	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑍)
12		dchrisum.b	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
14		elfzelz 13554	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℤ)
15	14	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℤ)
16	8, 9, 10, 11, 13, 15	dchrzrhcl 27377	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
17		elfznn 13583	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
18	17	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
19		mucl 27273	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
20	19	zred 12702	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℝ)
21		nndivre 12279	. . . . . . . . 9 ⊢ (((μ‘𝑑) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
22	20, 21	mpancom 700	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ → ((μ‘𝑑) / 𝑑) ∈ ℝ)
23	18, 22	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
24	23	recnd 11239	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
25	16, 24	mulcld 11231	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
26	7, 25	fsumcl 15786	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
27		dchrisumn0.t	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
28		climcl 15552	. . . . . 6 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ)
29	27, 28	syl 18	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℂ)
30	29	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
31	26, 30	mulcld 11231	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ)
32	1	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
33		subcl 11458	. . . . 5 ⊢ ((1 ∈ ℂ ∧ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ)
34	2, 31, 33	sylancr 598	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ)
35		1red 11211	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
36		dchrisumn0.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
37		elrege0 13483	. . . . . 6 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
38	36, 37	sylib 221	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
39	38	simpld 499	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
40		fzfid 14011	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
41	25	adantlrr 733	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
42		nnuz 12903	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
43		1zzd 12627	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
44	12	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
45		nnz 12614	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
46	45	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
47	8, 9, 10, 11, 44, 46	dchrzrhcl 27377	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
48		nncn 12243	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
49	48	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
50		nnne0 12272	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
51	50	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
52	47, 49, 51	divcld 11993	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
53		dchrisumn0.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
54		2fveq3 6889	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
55		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
56	54, 55	oveq12d 7431	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
57	56	cbvmptv 5219	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
58	53, 57	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
59	52, 58	fmptd 7112	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
60	59	ffvelcdmda 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
61	42, 43, 60	serf 14068	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
62	61	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → seq1( + , 𝐹):ℕ⟶ℂ)
63		simprl 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
64	63	rpred 13062	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
65		nndivre 12279	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
66	64, 17, 65	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
67	17	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
68	67	nncnd 12251	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
69	68	mullidd 11229	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
70		fznnfl 13897	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
71	64, 70	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
72	71	simplbda 504	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
73	69, 72	eqbrtrd 5137	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
74		1red 11211	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
75	64	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
76	67	nnrpd 13060	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
77	74, 75, 76	lemuldivd 13111	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
78	73, 77	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
79		flge1nn 13856	. . . . . . . . . . 11 ⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
80	66, 78, 79	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
81	62, 80	ffvelcdmd 7083	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ)
82	41, 81	mulcld 11231	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) ∈ ℂ)
83	29	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
84	41, 83	mulcld 11231	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ)
85	40, 82, 84	fsumsub 15841	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)))
86	41, 81, 83	subdid 11672	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) = ((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)))
87	86	sumeq2dv 15755	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)))
88	12	ad3antrrr 742	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑋 ∈ 𝐷)
89	14	ad2antlr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℤ)
90		elfzelz 13554	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℤ)
91	90	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℤ)
92	8, 9, 10, 11, 88, 89, 91	dchrzrhmul 27378	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘(𝑑 · 𝑚))) = ((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))))
93	92	oveq1d 7428	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚)))
94	16	adantlrr 733	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
95	94	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
96	68	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
97	8, 9, 10, 11, 88, 91	dchrzrhcl 27377	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
98		elfznn 13583	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
99	98	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
100	99	nncnd 12251	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
101	67	nnne0d 12288	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≠ 0)
102	101	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
103	99	nnne0d 12288	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ≠ 0)
104	95, 96, 97, 100, 102, 103	divmuldivd 12034	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚)))
105	93, 104	eqtr4d 2807	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
106	105	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))))
107	67, 19	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℤ)
108	107	zcnd 12703	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℂ)
109	108	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (μ‘𝑑) ∈ ℂ)
110	95, 96, 102	divcld 11993	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ)
111	97, 100, 103	divcld 11993	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
112	109, 110, 111	mulassd 11234	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))))
113	109, 95, 96, 102	div12d 12029	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) = ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)))
114	113	oveq1d 7428	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (((μ‘𝑑) · ((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
115	106, 112, 114	3eqtr2d 2810	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
116	115	sumeq2dv 15755	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
117		fzfid 14011	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
118		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝜑)
119	118, 98, 52	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
120	117, 41, 119	fsummulc2 15837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
121		ovex 7446	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
122	56, 53, 121	fvmpt 6992	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
123	99, 122	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
124	80, 42	eleqtrdi 2879	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ (ℤ_≥‘1))
125	123, 124, 119	fsumser 15783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
126	125	oveq2d 7429	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))))
127	116, 120, 126	3eqtr2rd 2811	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
128	127	sumeq2dv 15755	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
129		2fveq3 6889	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚))))
130		id 23	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑑 · 𝑚) → 𝑛 = (𝑑 · 𝑚))
131	129, 130	oveq12d 7431	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))
132	131	oveq2d 7429	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
133		elrabi 3655	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} → 𝑑 ∈ ℕ)
134	133	ad2antll 741	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
135	134, 19	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ)
136	135	zcnd 12703	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
137	12	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
138		elfzelz 13554	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
139	138	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
140	8, 9, 10, 11, 137, 139	dchrzrhcl 27377	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
141		fz1ssnn 13585	. . . . . . . . . . . . . . . . 17 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
142	141	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ⊆ ℕ)
143	142	sselda 3945	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
144	143	nncnd 12251	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
145	143	nnne0d 12288	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
146	140, 144, 145	divcld 11993	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
147	146	adantrr 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
148	136, 147	mulcld 11231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) ∈ ℂ)
149	132, 64, 148	dvdsflsumcom 27320	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
150		2fveq3 6889	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1)))
151		id 23	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → 𝑛 = 1)
152	150, 151	oveq12d 7431	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1))
153		simprr 784	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
154		flge1nn 13856	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
155	64, 153, 154	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
156	155, 42	eleqtrdi 2879	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
157		eluzfz1 13561	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝑥)))
158	156, 157	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ (1...(⌊‘𝑥)))
159	152, 40, 142, 158, 146	musumsum 27324	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((𝑋‘(𝐿‘1)) / 1))
160	128, 149, 159	3eqtr2d 2810	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = ((𝑋‘(𝐿‘1)) / 1))
161	8, 9, 10, 11, 12	dchrzrh1 27376	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1)
162	161	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑋‘(𝐿‘1)) = 1)
163	162	oveq1d 7428	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1))
164		1div1e1 11907	. . . . . . . . . 10 ⊢ (1 / 1) = 1
165	163, 164	eqtrdi 2820	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = 1)
166	160, 165	eqtr2d 2805	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))))
167	29	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 ∈ ℂ)
168	40, 167, 41	fsummulc1 15838	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))
169	166, 168	oveq12d 7431	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)))
170	85, 87, 169	3eqtr4rd 2815	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
171	170	fveq2d 6888	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) = (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))))
172	81, 83	subcld 11571	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ)
173	41, 172	mulcld 11231	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ)
174	40, 173	fsumcl 15786	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ)
175	174	abscld 15492	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ)
176	173	abscld 15492	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ)
177	40, 176	fsumrecl 15787	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ)
178	39	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℝ)
179	40, 173	fsumabs 15855	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))))
180		reflcl 13831	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
181	64, 180	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ)
182	181, 178	remulcld 11241	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ∈ ℝ)
183	182, 63	rerpdivcld 13093	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ∈ ℝ)
184	178, 63	rerpdivcld 13093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℝ)
185	184	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℝ)
186	41	abscld 15492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ)
187	67	nnrecred 12289	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℝ)
188	172	abscld 15492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ)
189	76	rpred 13062	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ)
190	185, 189	remulcld 11241	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℝ)
191	41	absge0d 15500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))))
192	172	absge0d 15500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
193	94	abscld 15492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ)
194	24	adantlrr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
195	194	abscld 15492	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ)
196	94	absge0d 15500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑))))
197	194	absge0d 15500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑑) / 𝑑)))
198		eqid 2769	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑍) = (Base‘𝑍)
199	12	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
200		rpvmasum.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ)
201	200	nnnn0d 12567	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
202	9, 198, 11	znzrhfo 21668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
203		fof 6795	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
204	201, 202, 203	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
205	204	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐿:ℤ⟶(Base‘𝑍))
206		ffvelcdm 7079	. . . . . . . . . . . . . . 15 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍))
207	205, 14, 206	syl2an 607	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑑) ∈ (Base‘𝑍))
208	8, 10, 9, 198, 199, 207	dchrabs2 27394	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1)
209	108, 68, 101	absdivd 15511	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑)))
210	76	rprege0d 13069	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
211		absid 15349	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ ℝ ∧ 0 ≤ 𝑑) → (abs‘𝑑) = 𝑑)
212	210, 211	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘𝑑) = 𝑑)
213	212	oveq2d 7429	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
214	209, 213	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
215	108	abscld 15492	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ∈ ℝ)
216		mule1 27280	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (abs‘(μ‘𝑑)) ≤ 1)
217	67, 216	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ≤ 1)
218	215, 74, 76, 217	lediv1dd 13120	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑))
219	214, 218	eqbrtrd 5137	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ≤ (1 / 𝑑))
220	193, 74, 195, 187, 196, 197, 208, 219	lemul12ad 12159	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑)))
221	94, 194	absmuld 15510	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) = ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))))
222	187	recnd 11239	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℂ)
223	222	mullidd 11229	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · (1 / 𝑑)) = (1 / 𝑑))
224	223	eqcomd 2775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) = (1 · (1 / 𝑑)))
225	220, 221, 224	3brtr4d 5147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑))
226		2fveq3 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
227	226	fvoveq1d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
228		oveq2 7421	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑)))
229	227, 228	breq12d 5126	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 / 𝑑) → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑))))
230		dchrisumn0.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))
231	230	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))
232		1re 11210	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
233		elicopnf 13474	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℝ → ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑))))
234	232, 233	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)))
235	66, 78, 234	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ (1[,)+∞))
236	229, 231, 235	rspcdva 3591	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))
237	178	recnd 11239	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℂ)
238	237	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
239		rpcnne0 13037	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
240	239	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
241	240	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
242		divdiv2 11929	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
243	238, 241, 68, 101, 242	syl112anc 1399	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
244		div23 11893	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
245	238, 68, 241, 244	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
246	243, 245	eqtrd 2804	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 / 𝑥) · 𝑑))
247	236, 246	breqtrd 5141	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ ((𝐶 / 𝑥) · 𝑑))
248	186, 187, 188, 190, 191, 192, 225, 247	lemul12ad 12159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
249	41, 172	absmuld 15510	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) = ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))))
250	184	recnd 11239	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℂ)
251	250	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℂ)
252	251, 68, 101	divcan4d 11999	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥))
253	251, 68	mulcld 11231	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℂ)
254	253, 68, 101	divrec2d 11997	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
255	252, 254	eqtr3d 2806	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
256	248, 249, 255	3brtr4d 5147	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (𝐶 / 𝑥))
257	40, 176, 185, 256	fsumle 15853	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥))
258	155	nnnn0d 12567	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ₀)
259		hashfz1 14384	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
260	258, 259	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
261	260	oveq1d 7428	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
262		fsumconst 15843	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
263	40, 250, 262	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
264	155	nncnd 12251	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℂ)
265		divass 11892	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
266	264, 237, 240, 265	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
267	261, 263, 266	3eqtr4d 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = (((⌊‘𝑥) · 𝐶) / 𝑥))
268	257, 267	breqtrd 5141	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (((⌊‘𝑥) · 𝐶) / 𝑥))
269	38	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
270		flle 13834	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
271	64, 270	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥)
272		lemul1a 12071	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (⌊‘𝑥) ≤ 𝑥) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
273	181, 64, 269, 271, 272	syl31anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
274	182, 178, 63	ledivmuld 13115	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶)))
275	273, 274	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶)
276	177, 183, 178, 268, 275	letrd 11369	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶)
277	175, 177, 178, 179, 276	letrd 11369	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶)
278	171, 277	eqbrtrd 5137	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ≤ 𝐶)
279	32, 34, 35, 39, 278	elo1d 15589	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1))
280	6, 31, 279	o1dif 15683	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1)))
281	5, 280	mpbid 235	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 {crab 3423 ⊆ wss 3913 class class class wbr 5113 ↦ cmpt 5196 ⟶wf 6535 –onto→wfo 6537 ‘cfv 6539 (class class class)co 7413 Fincfn 8945 ℂcc 11100 ℝcr 11101 0cc0 11102 1c1 11103 + caddc 11105 · cmul 11107 +∞cpnf 11242 ≤ cle 11246 − cmin 11443 / cdiv 11873 ℕcn 12235 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12864 ℝ⁺crp 13018 [,)cico 13376 ...cfz 13537 ⌊cfl 13825 seqcseq 14039 ♯chash 14368 abscabs 15287 ⇝ cli 15537 𝑂(1)co1 15539 Σcsu 15739 ∥ cdvds 16312 Basecbs 17271 0_gc0g 17494 ℤRHomczrh 21620 ℤ/nℤczn 21623 μcmu 27227 DChrcdchr 27364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-inf2 9612 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179 ax-pre-sup 11180 ax-addf 11181 ax-mulf 11182

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7677 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-ec 8698 df-qs 8702 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9324 df-fi 9373 df-sup 9404 df-inf 9405 df-oi 9474 df-dju 9889 df-card 9927 df-acn 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-div 11874 df-nn 12236 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12865 df-q 12975 df-rp 13019 df-xneg 13139 df-xadd 13140 df-xmul 13141 df-ioo 13378 df-ioc 13379 df-ico 13380 df-icc 13381 df-fz 13538 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-o1 15543 df-lo1 15544 df-sum 15740 df-ef 16123 df-sin 16125 df-cos 16126 df-pi 16128 df-dvds 16313 df-gcd 16555 df-prm 16732 df-pc 16899 df-struct 17209 df-sets 17226 df-slot 17244 df-ndx 17256 df-base 17272 df-ress 17293 df-plusg 17325 df-mulr 17326 df-starv 17327 df-sca 17328 df-vsca 17329 df-ip 17330 df-tset 17331 df-ple 17332 df-ds 17334 df-unif 17335 df-hom 17336 df-cco 17337 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17558 df-qtop 17563 df-imas 17564 df-qus 17565 df-xps 17566 df-mre 17640 df-mrc 17641 df-acs 17643 df-mgm 18700 df-sgrp 18779 df-mnd 18795 df-mhm 18843 df-submnd 18844 df-grp 19005 df-minusg 19006 df-sbg 19007 df-mulg 19136 df-subg 19191 df-nsg 19192 df-eqg 19193 df-ghm 19286 df-cntz 19389 df-od 19600 df-cmn 19854 df-abl 19855 df-mgp 20219 df-rng 20233 df-ur 20266 df-ring 20319 df-cring 20320 df-oppr 20421 df-dvdsr 20441 df-unit 20442 df-invr 20472 df-dvr 20485 df-rhm 20556 df-subrng 20633 df-subrg 20657 df-drng 20817 df-lmod 20963 df-lss 21033 df-lsp 21073 df-sra 21274 df-rgmod 21275 df-lidl 21312 df-rsp 21313 df-2idl 21362 df-psmet 21485 df-xmet 21486 df-met 21487 df-bl 21488 df-mopn 21489 df-fbas 21490 df-fg 21491 df-cnfld 21494 df-zring 21568 df-zrh 21624 df-zn 21627 df-top 23022 df-topon 23039 df-topsp 23061 df-bases 23074 df-cld 23147 df-ntr 23148 df-cls 23149 df-nei 23226 df-lp 23264 df-perf 23265 df-cn 23355 df-cnp 23356 df-haus 23443 df-tx 23690 df-hmeo 23883 df-fil 23974 df-fm 24066 df-flim 24067 df-flf 24068 df-xms 24448 df-ms 24449 df-tms 24450 df-cncf 25008 df-limc 25996 df-dv 25997 df-log 26689 df-cxp 26690 df-mu 27233 df-dchr 27365

This theorem is referenced by: dchrvmasumiflem2 27634 dchrmusumlem 27654

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