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Theorem dchrmusum2 25740

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 12235	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2		ax-1cn 10430	. . . 4 ⊢ 1 ∈ ℂ
3		o1const 14798	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
4	1, 2, 3	mp2an 688	. . 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 13179	. . . . 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 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
14		elfzelz 12747	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℤ)
15	14	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℤ)
16	8, 9, 10, 11, 13, 15	dchrzrhcl 25491	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
17		elfznn 12775	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
18	17	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
19		mucl 25388	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
20	19	zred 11925	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℝ)
21		nndivre 11515	. . . . . . . . 9 ⊢ (((μ‘𝑑) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
22	20, 21	mpancom 684	. . . . . . . 8 ⊢ (𝑑 ∈ ℕ → ((μ‘𝑑) / 𝑑) ∈ ℝ)
23	18, 22	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
24	23	recnd 10504	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
25	16, 24	mulcld 10496	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
26	7, 25	fsumcl 14911	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
27		dchrisumn0.t	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
28		climcl 14678	. . . . . 6 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ)
29	27, 28	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℂ)
30	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
31	26, 30	mulcld 10496	. . 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 10721	. . . . 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 587	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ)
35		1red 10477	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
36		dchrisumn0.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
37		elrege0 12681	. . . . . 6 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
38	36, 37	sylib 219	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
39	38	simpld 495	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
40		fzfid 13179	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
41	25	adantlrr 717	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
42		nnuz 12119	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
43		1zzd 11851	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
44	12	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
45		nnz 11842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
46	45	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
47	8, 9, 10, 11, 44, 46	dchrzrhcl 25491	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
48		nncn 11483	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
49	48	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
50		nnne0 11508	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
51	50	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
52	47, 49, 51	divcld 11253	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
53		dchrisumn0.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
54		2fveq3 6535	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
55		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
56	54, 55	oveq12d 7025	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
57	56	cbvmptv 5055	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
58	53, 57	eqtri 2817	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
59	52, 58	fmptd 6732	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
60	59	ffvelrnda 6707	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
61	42, 43, 60	serf 13236	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
62	61	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → seq1( + , 𝐹):ℕ⟶ℂ)
63		simprl 767	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
64	63	rpred 12270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
65		nndivre 11515	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
66	64, 17, 65	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
67	17	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
68	67	nncnd 11491	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
69	68	mulid2d 10494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
70		fznnfl 13068	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
71	64, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
72	71	simplbda 500	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
73	69, 72	eqbrtrd 4978	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
74		1red 10477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
75	64	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
76	67	nnrpd 12268	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
77	74, 75, 76	lemuldivd 12319	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
78	73, 77	mpbid 233	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
79		flge1nn 13029	. . . . . . . . . . 11 ⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
80	66, 78, 79	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
81	62, 80	ffvelrnd 6708	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ)
82	41, 81	mulcld 10496	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) ∈ ℂ)
83	29	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
84	41, 83	mulcld 10496	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ)
85	40, 82, 84	fsumsub 14964	. . . . . . 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 10933	. . . . . . . 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 14881	. . . . . . 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 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑋 ∈ 𝐷)
89	14	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℤ)
90		elfzelz 12747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℤ)
91	90	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℤ)
92	8, 9, 10, 11, 88, 89, 91	dchrzrhmul 25492	. . . . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
95	94	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
96	68	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
97	8, 9, 10, 11, 88, 91	dchrzrhcl 25491	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
98		elfznn 12775	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
99	98	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
100	99	nncnd 11491	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
101	67	nnne0d 11524	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≠ 0)
102	101	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
103	99	nnne0d 11524	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ≠ 0)
104	95, 96, 97, 100, 102, 103	divmuldivd 11294	. . . . . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . . 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 7023	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℤ)
108	107	zcnd 11926	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℂ)
109	108	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (μ‘𝑑) ∈ ℂ)
110	95, 96, 102	divcld 11253	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ)
111	97, 100, 103	divcld 11253	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
112	109, 110, 111	mulassd 10499	. . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . 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 7022	. . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . 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 14881	. . . . . . . . . . . 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 13179	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
118		simpll 763	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝜑)
119	118, 98, 52	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
120	117, 41, 119	fsummulc2 14960	. . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
122	56, 53, 121	fvmpt 6626	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
123	99, 122	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
124	80, 42	syl6eleq 2891	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ (ℤ_≥‘1))
125	123, 124, 119	fsumser 14908	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
126	125	oveq2d 7023	. . . . . . . . . . . 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 2836	. . . . . . . . . . 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 14881	. . . . . . . . . 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 6535	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚))))
130		id 22	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑑 · 𝑚) → 𝑛 = (𝑑 · 𝑚))
131	129, 130	oveq12d 7025	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))
132	131	oveq2d 7023	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
133		elrabi 3608	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} → 𝑑 ∈ ℕ)
134	133	ad2antll 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
135	134, 19	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ)
136	135	zcnd 11926	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
137	12	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
138		elfzelz 12747	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
139	138	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
140	8, 9, 10, 11, 137, 139	dchrzrhcl 25491	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
141		fz1ssnn 12777	. . . . . . . . . . . . . . . . 17 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
142	141	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ⊆ ℕ)
143	142	sselda 3884	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
144	143	nncnd 11491	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
145	143	nnne0d 11524	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
146	140, 144, 145	divcld 11253	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
147	146	adantrr 713	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
148	136, 147	mulcld 10496	. . . . . . . . . . 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 25435	. . . . . . . . . 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 6535	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1)))
151		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → 𝑛 = 1)
152	150, 151	oveq12d 7025	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1))
153		simprr 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
154		flge1nn 13029	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
155	64, 153, 154	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
156	155, 42	syl6eleq 2891	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
157		eluzfz1 12753	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ (ℤ_≥‘1) → 1 ∈ (1...(⌊‘𝑥)))
158	156, 157	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ (1...(⌊‘𝑥)))
159	152, 40, 142, 158, 146	musumsum 25439	. . . . . . . . . 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 2835	. . . . . . . . 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 25490	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1)
162	161	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑋‘(𝐿‘1)) = 1)
163	162	oveq1d 7022	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1))
164		1div1e1 11167	. . . . . . . . . 10 ⊢ (1 / 1) = 1
165	163, 164	syl6eq 2845	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = 1)
166	160, 165	eqtr2d 2830	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))))
167	29	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 ∈ ℂ)
168	40, 167, 41	fsummulc1 14961	. . . . . . . 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 7025	. . . . . . 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 2840	. . . . . 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 6534	. . . . 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 10834	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ)
173	41, 172	mulcld 10496	. . . . . . . 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 14911	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ)
175	174	abscld 14618	. . . . . 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 14618	. . . . . . 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 14912	. . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℝ)
179	40, 173	fsumabs 14977	. . . . . 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 13004	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
181	64, 180	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ)
182	181, 178	remulcld 10506	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ∈ ℝ)
183	182, 63	rerpdivcld 12301	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ∈ ℝ)
184	178, 63	rerpdivcld 12301	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℝ)
185	184	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℝ)
186	41	abscld 14618	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ)
187	67	nnrecred 11525	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℝ)
188	172	abscld 14618	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ)
189	76	rpred 12270	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ)
190	185, 189	remulcld 10506	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℝ)
191	41	absge0d 14626	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))))
192	172	absge0d 14626	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
193	94	abscld 14618	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ)
194	24	adantlrr 717	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
195	194	abscld 14618	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ)
196	94	absge0d 14626	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑))))
197	194	absge0d 14626	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑑) / 𝑑)))
198		eqid 2793	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑍) = (Base‘𝑍)
199	12	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
200		rpvmasum.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ)
201	200	nnnn0d 11792	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
202	9, 198, 11	znzrhfo 20364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
203		fof 6450	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
204	201, 202, 203	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
205	204	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐿:ℤ⟶(Base‘𝑍))
206		ffvelrn 6705	. . . . . . . . . . . . . . 15 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍))
207	205, 14, 206	syl2an 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑑) ∈ (Base‘𝑍))
208	8, 10, 9, 198, 199, 207	dchrabs2 25508	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1)
209	108, 68, 101	absdivd 14637	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑)))
210	76	rprege0d 12277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
211		absid 14478	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑 ∈ ℝ ∧ 0 ≤ 𝑑) → (abs‘𝑑) = 𝑑)
212	210, 211	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘𝑑) = 𝑑)
213	212	oveq2d 7023	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
214	209, 213	eqtrd 2829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
215	108	abscld 14618	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ∈ ℝ)
216		mule1 25395	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (abs‘(μ‘𝑑)) ≤ 1)
217	67, 216	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ≤ 1)
218	215, 74, 76, 217	lediv1dd 12328	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑))
219	214, 218	eqbrtrd 4978	. . . . . . . . . . . . 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 11419	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑)))
221	94, 194	absmuld 14636	. . . . . . . . . . . 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 10504	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℂ)
223	222	mulid2d 10494	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · (1 / 𝑑)) = (1 / 𝑑))
224	223	eqcomd 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) = (1 · (1 / 𝑑)))
225	220, 221, 224	3brtr4d 4988	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑))
226		2fveq3 6535	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
227	226	fvoveq1d 7029	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
228		oveq2 7015	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑)))
229	227, 228	breq12d 4969	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))
232		1re 10476	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
233		elicopnf 12672	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ (1[,)+∞))
236	229, 231, 235	rspcdva 3560	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))
237	178	recnd 10504	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℂ)
238	237	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
239		rpcnne0 12246	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
240	239	ad2antrl 724	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
241	240	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
242		divdiv2 11189	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
243	238, 241, 68, 101, 242	syl112anc 1365	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
244		div23 11154	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
245	238, 68, 241, 244	syl3anc 1362	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
246	243, 245	eqtrd 2829	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 / 𝑥) · 𝑑))
247	236, 246	breqtrd 4982	. . . . . . . . . . 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 11419	. . . . . . . . . 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 14636	. . . . . . . . . 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 10504	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℂ)
251	250	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℂ)
252	251, 68, 101	divcan4d 11259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥))
253	251, 68	mulcld 10496	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℂ)
254	253, 68, 101	divrec2d 11257	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
255	252, 254	eqtr3d 2831	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
256	248, 249, 255	3brtr4d 4988	. . . . . . . . 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 14975	. . . . . . . 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 11792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ₀)
259		hashfz1 13544	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
260	258, 259	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
261	260	oveq1d 7022	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
262		fsumconst 14966	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
263	40, 250, 262	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
264	155	nncnd 11491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℂ)
265		divass 11153	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
266	264, 237, 240, 265	syl3anc 1362	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
267	261, 263, 266	3eqtr4d 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = (((⌊‘𝑥) · 𝐶) / 𝑥))
268	257, 267	breqtrd 4982	. . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
270		flle 13007	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
271	64, 270	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥)
272		lemul1a 11331	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (⌊‘𝑥) ≤ 𝑥) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
273	181, 64, 269, 271, 272	syl31anc 1364	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
274	182, 178, 63	ledivmuld 12323	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶)))
275	273, 274	mpbird 258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶)
276	177, 183, 178, 268, 275	letrd 10633	. . . . . 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 10633	. . . . 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 4978	. . . 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 14715	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1))
280	6, 31, 279	o1dif 14808	. 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 233	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 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ≠ wne 2982 ∀wral 3103 {crab 3107 ⊆ wss 3854 class class class wbr 4956 ↦ cmpt 5035 ⟶wf 6213 –onto→wfo 6215 ‘cfv 6217 (class class class)co 7007 Fincfn 8347 ℂcc 10370 ℝcr 10371 0cc0 10372 1c1 10373 + caddc 10375 · cmul 10377 +∞cpnf 10507 ≤ cle 10511 − cmin 10706 / cdiv 11134 ℕcn 11475 ℕ₀cn0 11734 ℤcz 11818 ℤ_≥cuz 12082 ℝ⁺crp 12228 [,)cico 12579 ...cfz 12731 ⌊cfl 12998 seqcseq 13207 ♯chash 13528 abscabs 14415 ⇝ cli 14663 𝑂(1)co1 14665 Σcsu 14864 ∥ cdvds 15428 Basecbs 16300 0_gc0g 16530 ℤRHomczrh 20317 ℤ/nℤczn 20320 μcmu 25342 DChrcdchr 25478

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-inf2 8939 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 ax-addf 10451 ax-mulf 10452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-fal 1533 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-iin 4822 df-disj 4925 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-se 5395 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-isom 6226 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-of 7258 df-om 7428 df-1st 7536 df-2nd 7537 df-supp 7673 df-tpos 7734 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-2o 7945 df-oadd 7948 df-omul 7949 df-er 8130 df-ec 8132 df-qs 8136 df-map 8249 df-pm 8250 df-ixp 8301 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-fsupp 8670 df-fi 8711 df-sup 8742 df-inf 8743 df-oi 8810 df-dju 9165 df-card 9203 df-acn 9206 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-2 11537 df-3 11538 df-4 11539 df-5 11540 df-6 11541 df-7 11542 df-8 11543 df-9 11544 df-n0 11735 df-xnn0 11805 df-z 11819 df-dec 11937 df-uz 12083 df-q 12187 df-rp 12229 df-xneg 12346 df-xadd 12347 df-xmul 12348 df-ioo 12581 df-ioc 12582 df-ico 12583 df-icc 12584 df-fz 12732 df-fzo 12873 df-fl 13000 df-mod 13076 df-seq 13208 df-exp 13268 df-fac 13472 df-bc 13501 df-hash 13529 df-shft 14248 df-cj 14280 df-re 14281 df-im 14282 df-sqrt 14416 df-abs 14417 df-limsup 14650 df-clim 14667 df-rlim 14668 df-o1 14669 df-lo1 14670 df-sum 14865 df-ef 15242 df-sin 15244 df-cos 15245 df-pi 15247 df-dvds 15429 df-gcd 15665 df-prm 15833 df-pc 15991 df-struct 16302 df-ndx 16303 df-slot 16304 df-base 16306 df-sets 16307 df-ress 16308 df-plusg 16395 df-mulr 16396 df-starv 16397 df-sca 16398 df-vsca 16399 df-ip 16400 df-tset 16401 df-ple 16402 df-ds 16404 df-unif 16405 df-hom 16406 df-cco 16407 df-rest 16513 df-topn 16514 df-0g 16532 df-gsum 16533 df-topgen 16534 df-pt 16535 df-prds 16538 df-xrs 16592 df-qtop 16597 df-imas 16598 df-qus 16599 df-xps 16600 df-mre 16674 df-mrc 16675 df-acs 16677 df-mgm 17669 df-sgrp 17711 df-mnd 17722 df-mhm 17762 df-submnd 17763 df-grp 17852 df-minusg 17853 df-sbg 17854 df-mulg 17970 df-subg 18018 df-nsg 18019 df-eqg 18020 df-ghm 18085 df-cntz 18176 df-od 18375 df-cmn 18623 df-abl 18624 df-mgp 18918 df-ur 18930 df-ring 18977 df-cring 18978 df-oppr 19051 df-dvdsr 19069 df-unit 19070 df-invr 19100 df-dvr 19111 df-rnghom 19145 df-drng 19182 df-subrg 19211 df-lmod 19314 df-lss 19382 df-lsp 19422 df-sra 19622 df-rgmod 19623 df-lidl 19624 df-rsp 19625 df-2idl 19682 df-psmet 20207 df-xmet 20208 df-met 20209 df-bl 20210 df-mopn 20211 df-fbas 20212 df-fg 20213 df-cnfld 20216 df-zring 20288 df-zrh 20321 df-zn 20324 df-top 21174 df-topon 21191 df-topsp 21213 df-bases 21226 df-cld 21299 df-ntr 21300 df-cls 21301 df-nei 21378 df-lp 21416 df-perf 21417 df-cn 21507 df-cnp 21508 df-haus 21595 df-tx 21842 df-hmeo 22035 df-fil 22126 df-fm 22218 df-flim 22219 df-flf 22220 df-xms 22601 df-ms 22602 df-tms 22603 df-cncf 23157 df-limc 24135 df-dv 24136 df-log 24809 df-cxp 24810 df-mu 25348 df-dchr 25479

This theorem is referenced by: dchrvmasumiflem2 25748 dchrmusumlem 25768

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