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Theorem dchrmusum2 26069

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 12395	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
2		ax-1cn 10594	. . . 4 ⊢ 1 ∈ ℂ
3		o1const 14975	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 1) ∈ 𝑂(1))
4	1, 2, 3	mp2an 690	. . 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 13340	. . . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
14		elfzelz 12907	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℤ)
15	14	adantl 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℤ)
16	8, 9, 10, 11, 13, 15	dchrzrhcl 25820	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
17		elfznn 12935	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
18	17	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
19		mucl 25717	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
20	19	zred 12086	. . . . . . . . 9 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℝ)
21		nndivre 11677	. . . . . . . . 9 ⊢ (((μ‘𝑑) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
22	20, 21	mpancom 686	. . . . . . . 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 10668	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
25	16, 24	mulcld 10660	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
26	7, 25	fsumcl 15089	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
27		dchrisumn0.t	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
28		climcl 14855	. . . . . 6 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ)
29	27, 28	syl 17	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ ℂ)
30	29	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
31	26, 30	mulcld 10660	. . 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 10884	. . . . 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 589	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ)
35		1red 10641	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
36		dchrisumn0.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
37		elrege0 12841	. . . . . 6 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
38	36, 37	sylib 220	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
39	38	simpld 497	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ)
40		fzfid 13340	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
41	25	adantlrr 719	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
42		nnuz 12280	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
43		1zzd 12012	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
44	12	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
45		nnz 12003	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
46	45	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
47	8, 9, 10, 11, 44, 46	dchrzrhcl 25820	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
48		nncn 11645	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
49	48	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
50		nnne0 11670	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
51	50	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
52	47, 49, 51	divcld 11415	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
53		dchrisumn0.f	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
54		2fveq3 6674	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
55		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
56	54, 55	oveq12d 7173	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
57	56	cbvmptv 5168	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
58	53, 57	eqtri 2844	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
59	52, 58	fmptd 6877	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
60	59	ffvelrnda 6850	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
61	42, 43, 60	serf 13397	. . . . . . . . . . 11 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
62	61	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → seq1( + , 𝐹):ℕ⟶ℂ)
63		simprl 769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
64	63	rpred 12430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
65		nndivre 11677	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
66	64, 17, 65	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
67	17	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
68	67	nncnd 11653	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℂ)
69	68	mulid2d 10658	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) = 𝑑)
70		fznnfl 13229	. . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
73	69, 72	eqbrtrd 5087	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · 𝑑) ≤ 𝑥)
74		1red 10641	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
75	64	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
76	67	nnrpd 12428	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
77	74, 75, 76	lemuldivd 12479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑑) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑑)))
78	73, 77	mpbid 234	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑑))
79		flge1nn 13190	. . . . . . . . . . 11 ⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
80	66, 78, 79	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ)
81	62, 80	ffvelrnd 6851	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ)
82	41, 81	mulcld 10660	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) ∈ ℂ)
83	29	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
84	41, 83	mulcld 10660	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ)
85	40, 82, 84	fsumsub 15142	. . . . . . 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 11095	. . . . . . . 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 15059	. . . . . . 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 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑋 ∈ 𝐷)
89	14	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℤ)
90		elfzelz 12907	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℤ)
91	90	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℤ)
92	8, 9, 10, 11, 88, 89, 91	dchrzrhmul 25821	. . . . . . . . . . . . . . . . 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 7170	. . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
95	94	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
96	68	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
97	8, 9, 10, 11, 88, 91	dchrzrhcl 25820	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
98		elfznn 12935	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
99	98	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
100	99	nncnd 11653	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
101	67	nnne0d 11686	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≠ 0)
102	101	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
103	99	nnne0d 11686	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ≠ 0)
104	95, 96, 97, 100, 102, 103	divmuldivd 11456	. . . . . . . . . . . . . . . 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 2859	. . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . 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 12087	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℂ)
109	108	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (μ‘𝑑) ∈ ℂ)
110	95, 96, 102	divcld 11415	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ)
111	97, 100, 103	divcld 11415	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
112	109, 110, 111	mulassd 10663	. . . . . . . . . . . . . 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 11451	. . . . . . . . . . . . . . 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 7170	. . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . 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 15059	. . . . . . . . . . . 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 13340	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
118		simpll 765	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝜑)
119	118, 98, 52	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
120	117, 41, 119	fsummulc2 15138	. . . . . . . . . . . 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 7188	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
122	56, 53, 121	fvmpt 6767	. . . . . . . . . . . . . . 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	eleqtrdi 2923	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑑)) ∈ (ℤ_≥‘1))
125	123, 124, 119	fsumser 15086	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
126	125	oveq2d 7171	. . . . . . . . . . . 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 2863	. . . . . . . . . . 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 15059	. . . . . . . . . 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 6674	. . . . . . . . . . . . 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 7173	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))
132	131	oveq2d 7171	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))))
133		elrabi 3674	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} → 𝑑 ∈ ℕ)
134	133	ad2antll 727	. . . . . . . . . . . . . 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 12087	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
137	12	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
138		elfzelz 12907	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
139	138	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
140	8, 9, 10, 11, 137, 139	dchrzrhcl 25820	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
141		fz1ssnn 12937	. . . . . . . . . . . . . . . . 17 ⊢ (1...(⌊‘𝑥)) ⊆ ℕ
142	141	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ⊆ ℕ)
143	142	sselda 3966	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
144	143	nncnd 11653	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
145	143	nnne0d 11686	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
146	140, 144, 145	divcld 11415	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
147	146	adantrr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ)
148	136, 147	mulcld 10660	. . . . . . . . . . 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 25764	. . . . . . . . . 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 6674	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1)))
151		id 22	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → 𝑛 = 1)
152	150, 151	oveq12d 7173	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1))
153		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
154		flge1nn 13190	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
155	64, 153, 154	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
156	155, 42	eleqtrdi 2923	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
157		eluzfz1 12913	. . . . . . . . . . . 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 25768	. . . . . . . . . 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 2862	. . . . . . . . 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 25819	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1)
162	161	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑋‘(𝐿‘1)) = 1)
163	162	oveq1d 7170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1))
164		1div1e1 11329	. . . . . . . . . 10 ⊢ (1 / 1) = 1
165	163, 164	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = 1)
166	160, 165	eqtr2d 2857	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))))
167	29	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 ∈ ℂ)
168	40, 167, 41	fsummulc1 15139	. . . . . . . 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 7173	. . . . . . 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 2867	. . . . . 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 6673	. . . . 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 10996	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ)
173	41, 172	mulcld 10660	. . . . . . . 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 15089	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ)
175	174	abscld 14795	. . . . . 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 14795	. . . . . . 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 15090	. . . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℝ)
179	40, 173	fsumabs 15155	. . . . . 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 13165	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
181	64, 180	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℝ)
182	181, 178	remulcld 10670	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ∈ ℝ)
183	182, 63	rerpdivcld 12461	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ∈ ℝ)
184	178, 63	rerpdivcld 12461	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℝ)
185	184	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℝ)
186	41	abscld 14795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ)
187	67	nnrecred 11687	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℝ)
188	172	abscld 14795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ)
189	76	rpred 12430	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ)
190	185, 189	remulcld 10670	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℝ)
191	41	absge0d 14803	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))))
192	172	absge0d 14803	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
193	94	abscld 14795	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ)
194	24	adantlrr 719	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
195	194	abscld 14795	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ)
196	94	absge0d 14803	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑))))
197	194	absge0d 14803	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((μ‘𝑑) / 𝑑)))
198		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑍) = (Base‘𝑍)
199	12	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
200		rpvmasum.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ)
201	200	nnnn0d 11954	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
202	9, 198, 11	znzrhfo 20693	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
203		fof 6589	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐿:ℤ⟶(Base‘𝑍))
206		ffvelrn 6848	. . . . . . . . . . . . . . 15 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍))
207	205, 14, 206	syl2an 597	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑑) ∈ (Base‘𝑍))
208	8, 10, 9, 198, 199, 207	dchrabs2 25837	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1)
209	108, 68, 101	absdivd 14814	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑)))
210	76	rprege0d 12437	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
211		absid 14655	. . . . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
214	209, 213	eqtrd 2856	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑))
215	108	abscld 14795	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑑)) ∈ ℝ)
216		mule1 25724	. . . . . . . . . . . . . . . 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 12488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑))
219	214, 218	eqbrtrd 5087	. . . . . . . . . . . . 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 11581	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑)))
221	94, 194	absmuld 14813	. . . . . . . . . . . 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 10668	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) ∈ ℂ)
223	222	mulid2d 10658	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 · (1 / 𝑑)) = (1 / 𝑑))
224	223	eqcomd 2827	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1 / 𝑑) = (1 · (1 / 𝑑)))
225	220, 221, 224	3brtr4d 5097	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑))
226		2fveq3 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))
227	226	fvoveq1d 7177	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))
228		oveq2 7163	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑)))
229	227, 228	breq12d 5078	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))
232		1re 10640	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℝ
233		elicopnf 12832	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ (1[,)+∞))
236	229, 231, 235	rspcdva 3624	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))
237	178	recnd 10668	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐶 ∈ ℂ)
238	237	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
239		rpcnne0 12406	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
240	239	ad2antrl 726	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
241	240	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
242		divdiv2 11351	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
243	238, 241, 68, 101, 242	syl112anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥))
244		div23 11316	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
245	238, 68, 241, 244	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑))
246	243, 245	eqtrd 2856	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 / 𝑥) · 𝑑))
247	236, 246	breqtrd 5091	. . . . . . . . . . 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 11581	. . . . . . . . . 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 14813	. . . . . . . . . 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 10668	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 / 𝑥) ∈ ℂ)
251	250	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) ∈ ℂ)
252	251, 68, 101	divcan4d 11421	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥))
253	251, 68	mulcld 10660	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝐶 / 𝑥) · 𝑑) ∈ ℂ)
254	253, 68, 101	divrec2d 11419	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝐶 / 𝑥) · 𝑑) / 𝑑) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
255	252, 254	eqtr3d 2858	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / 𝑥) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑)))
256	248, 249, 255	3brtr4d 5097	. . . . . . . . 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 15153	. . . . . . . 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 11954	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ₀)
259		hashfz1 13705	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
260	258, 259	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
261	260	oveq1d 7170	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
262		fsumconst 15144	. . . . . . . . . 10 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
263	40, 250, 262	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)))
264	155	nncnd 11653	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℂ)
265		divass 11315	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
266	264, 237, 240, 265	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥)))
267	261, 263, 266	3eqtr4d 2866	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥) = (((⌊‘𝑥) · 𝐶) / 𝑥))
268	257, 267	breqtrd 5091	. . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
270		flle 13168	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
271	64, 270	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ≤ 𝑥)
272		lemul1a 11493	. . . . . . . . 9 ⊢ ((((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (⌊‘𝑥) ≤ 𝑥) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
273	181, 64, 269, 271, 272	syl31anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))
274	182, 178, 63	ledivmuld 12483	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶)))
275	273, 274	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (((⌊‘𝑥) · 𝐶) / 𝑥) ≤ 𝐶)
276	177, 183, 178, 268, 275	letrd 10796	. . . . . 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 10796	. . . . 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 5087	. . . 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 14892	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 − (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1))
280	6, 31, 279	o1dif 14985	. 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 234	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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 {crab 3142 ⊆ wss 3935 class class class wbr 5065 ↦ cmpt 5145 ⟶wf 6350 –onto→wfo 6352 ‘cfv 6354 (class class class)co 7155 Fincfn 8508 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 · cmul 10541 +∞cpnf 10671 ≤ cle 10675 − cmin 10869 / cdiv 11296 ℕcn 11637 ℕ₀cn0 11896 ℤcz 11980 ℤ_≥cuz 12242 ℝ⁺crp 12388 [,)cico 12739 ...cfz 12891 ⌊cfl 13159 seqcseq 13368 ♯chash 13689 abscabs 14592 ⇝ cli 14840 𝑂(1)co1 14842 Σcsu 15041 ∥ cdvds 15606 Basecbs 16482 0_gc0g 16712 ℤRHomczrh 20646 ℤ/nℤczn 20649 μcmu 25671 DChrcdchr 25807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-omul 8106 df-er 8288 df-ec 8290 df-qs 8294 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-acn 9370 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-o1 14846 df-lo1 14847 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-dvds 15607 df-gcd 15843 df-prm 16015 df-pc 16173 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-qus 16781 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-nsg 18276 df-eqg 18277 df-ghm 18355 df-cntz 18446 df-od 18655 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-rnghom 19466 df-drng 19503 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-sra 19943 df-rgmod 19944 df-lidl 19945 df-rsp 19946 df-2idl 20004 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-zn 20653 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-limc 24463 df-dv 24464 df-log 25139 df-cxp 25140 df-mu 25677 df-dchr 25808

This theorem is referenced by: dchrvmasumiflem2 26077 dchrmusumlem 26097

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