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Theorem dchrvmasumiflem2 27422

Description: Lemma for dchrvmasum 27445. (Contributed by Mario Carneiro, 5-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 )
dchrvmasumif.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrvmasumif.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrvmasumif.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrvmasumif.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
dchrvmasumif.g	⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))
dchrvmasumif.e	⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
dchrvmasumif.t	⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
dchrvmasumif.2	⊢ (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))

**Assertion**
Ref	Expression
dchrvmasumiflem2	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))

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

Proof of Theorem dchrvmasumiflem2 Dummy variables 𝑘 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11237	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		fzfid 13962	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
3		rpvmasum.g	. . . . . . . 8 ⊢ 𝐺 = (DChr‘𝑁)
4		rpvmasum.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
5		rpvmasum.d	. . . . . . . 8 ⊢ 𝐷 = (Base‘𝐺)
6		rpvmasum.l	. . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍)
7		dchrisum.b	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8	7	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
9		elfzelz 13525	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℤ)
10	9	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℤ)
11	3, 4, 5, 6, 8, 10	dchrzrhcl 27165	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ)
12		elfznn 13554	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
13	12	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
14		mucl 27060	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℤ)
16	15	zred 12688	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (μ‘𝑑) ∈ ℝ)
17	16, 13	nndivred 12288	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℝ)
18	17	recnd 11264	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑑) / 𝑑) ∈ ℂ)
19	11, 18	mulcld 11256	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
20	2, 19	fsumcl 15703	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ)
21		dchrvmasumif.s	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
22		climcl 15467	. . . . . . 7 ⊢ (seq1( + , 𝐹) ⇝ 𝑆 → 𝑆 ∈ ℂ)
23	21, 22	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑆 ∈ ℂ)
25	20, 24	mulcld 11256	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) ∈ ℂ)
26		0cnd 11229	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 = 0) → 0 ∈ ℂ)
27		df-ne 2936	. . . . . . 7 ⊢ (𝑆 ≠ 0 ↔ ¬ 𝑆 = 0)
28		dchrvmasumif.t	. . . . . . . . . 10 ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
29		climcl 15467	. . . . . . . . . 10 ⊢ (seq1( + , 𝐾) ⇝ 𝑇 → 𝑇 ∈ ℂ)
30	28, 29	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ℂ)
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑇 ∈ ℂ)
32	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑆 ∈ ℂ)
33		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑆 ≠ 0)
34	31, 32, 33	divcld 12012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑇 / 𝑆) ∈ ℂ)
35	27, 34	sylan2br 594	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑆 = 0) → (𝑇 / 𝑆) ∈ ℂ)
36	26, 35	ifclda 4559	. . . . 5 ⊢ (𝜑 → if(𝑆 = 0, 0, (𝑇 / 𝑆)) ∈ ℂ)
37	36	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → if(𝑆 = 0, 0, (𝑇 / 𝑆)) ∈ ℂ)
38		rpvmasum.a	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
39		rpvmasum.1	. . . . 5 ⊢ 1 = (0_g‘𝐺)
40		dchrisum.n1	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 1 )
41		dchrvmasumif.f	. . . . 5 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
42		dchrvmasumif.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
43		dchrvmasumif.1	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
44	4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43	dchrmusum2 27414	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆)) ∈ 𝑂(1))
45		rpssre 13005	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
46		o1const 15588	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ if(𝑆 = 0, 0, (𝑇 / 𝑆)) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ if(𝑆 = 0, 0, (𝑇 / 𝑆))) ∈ 𝑂(1))
47	45, 36, 46	sylancr 586	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ if(𝑆 = 0, 0, (𝑇 / 𝑆))) ∈ 𝑂(1))
48	25, 37, 44, 47	o1mul2 15593	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆)))) ∈ 𝑂(1))
49		fzfid 13962	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
50	8	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑋 ∈ 𝐷)
51		elfzelz 13525	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑘 ∈ ℤ)
52	51	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑘 ∈ ℤ)
53	3, 4, 5, 6, 50, 52	dchrzrhcl 27165	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
54		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
55	12	nnrpd 13038	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
56		rpdivcl 13023	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
57	54, 55, 56	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
58		elfznn 13554	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑘 ∈ ℕ)
59	58	nnrpd 13038	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑘 ∈ ℝ⁺)
60		ifcl 4569	. . . . . . . . . . . 12 ⊢ (((𝑥 / 𝑑) ∈ ℝ⁺ ∧ 𝑘 ∈ ℝ⁺) → if(𝑆 = 0, (𝑥 / 𝑑), 𝑘) ∈ ℝ⁺)
61	57, 59, 60	syl2an 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → if(𝑆 = 0, (𝑥 / 𝑑), 𝑘) ∈ ℝ⁺)
62	61	relogcld 26544	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) ∈ ℝ)
63	58	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑘 ∈ ℕ)
64	62, 63	nndivred 12288	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘) ∈ ℝ)
65	64	recnd 11264	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘) ∈ ℂ)
66	53, 65	mulcld 11256	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) ∈ ℂ)
67	49, 66	fsumcl 15703	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) ∈ ℂ)
68	19, 67	mulcld 11256	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) ∈ ℂ)
69	2, 68	fsumcl 15703	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) ∈ ℂ)
70	25, 37	mulcld 11256	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆))) ∈ ℂ)
71		0cn 11228	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
72	30	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
73		ifcl 4569	. . . . . . . . . 10 ⊢ ((0 ∈ ℂ ∧ 𝑇 ∈ ℂ) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
74	71, 72, 73	sylancr 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
75	19, 67, 74	subdid 11692	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = ((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇))))
76	75	sumeq2dv 15673	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇))))
77	19, 74	mulcld 11256	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
78	2, 68, 77	fsumsub 15758	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇))))
79	20, 24, 37	mulassd 11259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝑆 · if(𝑆 = 0, 0, (𝑇 / 𝑆)))))
80		ovif2 7513	. . . . . . . . . . . 12 ⊢ (𝑆 · if(𝑆 = 0, 0, (𝑇 / 𝑆))) = if(𝑆 = 0, (𝑆 · 0), (𝑆 · (𝑇 / 𝑆)))
81	23	mul01d 11435	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 · 0) = 0)
82	81	ifeq1d 4543	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑆 = 0, (𝑆 · 0), (𝑆 · (𝑇 / 𝑆))) = if(𝑆 = 0, 0, (𝑆 · (𝑇 / 𝑆))))
83	31, 32, 33	divcan2d 12014	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑆 · (𝑇 / 𝑆)) = 𝑇)
84	27, 83	sylan2br 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ¬ 𝑆 = 0) → (𝑆 · (𝑇 / 𝑆)) = 𝑇)
85	84	ifeq2da 4556	. . . . . . . . . . . . 13 ⊢ (𝜑 → if(𝑆 = 0, 0, (𝑆 · (𝑇 / 𝑆))) = if(𝑆 = 0, 0, 𝑇))
86	82, 85	eqtrd 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → if(𝑆 = 0, (𝑆 · 0), (𝑆 · (𝑇 / 𝑆))) = if(𝑆 = 0, 0, 𝑇))
87	80, 86	eqtrid 2779	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 · if(𝑆 = 0, 0, (𝑇 / 𝑆))) = if(𝑆 = 0, 0, 𝑇))
88	87	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑆 · if(𝑆 = 0, 0, (𝑇 / 𝑆))) = if(𝑆 = 0, 0, 𝑇))
89	88	oveq2d 7430	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (𝑆 · if(𝑆 = 0, 0, (𝑇 / 𝑆)))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇)))
90	71, 30, 73	sylancr 586	. . . . . . . . . . 11 ⊢ (𝜑 → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
91	90	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
92	2, 91, 19	fsummulc1 15755	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇)))
93	79, 89, 92	3eqtrrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇)) = ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆))))
94	93	oveq2d 7430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · if(𝑆 = 0, 0, 𝑇))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆)))))
95	76, 78, 94	3eqtrd 2771	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆)))))
96	95	mpteq2dva 5242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆))))))
97		dchrvmasumif.g	. . . . . 6 ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))
98		dchrvmasumif.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
99		dchrvmasumif.2	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))
100	4, 6, 38, 3, 5, 39, 7, 40, 41, 42, 21, 43, 97, 98, 28, 99	dchrvmasumiflem1 27421	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1))
101	96, 100	eqeltrrd 2829	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))) − ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆))))) ∈ 𝑂(1))
102	69, 70, 101	o1dif 15598	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑆) · if(𝑆 = 0, 0, (𝑇 / 𝑆)))) ∈ 𝑂(1)))
103	48, 102	mpbird 257	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))) ∈ 𝑂(1))
104	7	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
105		elfzelz 13525	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
106	105	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
107	3, 4, 5, 6, 104, 106	dchrzrhcl 27165	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
108		elfznn 13554	. . . . . . . 8 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
109	108	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
110		vmacl 27037	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
111		nndivre 12275	. . . . . . . 8 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
112	110, 111	mpancom 687	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
113	109, 112	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
114	113	recnd 11264	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
115	107, 114	mulcld 11256	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
116	2, 115	fsumcl 15703	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ)
117		relogcl 26496	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
118	117	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
119	118	recnd 11264	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
120		ifcl 4569	. . . 4 ⊢ (((log‘𝑥) ∈ ℂ ∧ 0 ∈ ℂ) → if(𝑆 = 0, (log‘𝑥), 0) ∈ ℂ)
121	119, 71, 120	sylancl 585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → if(𝑆 = 0, (log‘𝑥), 0) ∈ ℂ)
122	116, 121	addcld 11255	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0)) ∈ ℂ)
123	122	abscld 15407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ ℝ)
124	123	adantrr 716	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ ℝ)
125	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ ℕ)
126	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑋 ∈ 𝐷)
127	40	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑋 ≠ 1 )
128		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ⁺)
129		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
130	4, 6, 125, 3, 5, 39, 126, 127, 128, 129	dchrvmasum2if 27417	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))))
131	130	fveq2d 6895	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) = (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))))
132	124, 131	eqled 11339	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ≤ (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))))
133	1, 103, 69, 122, 132	o1le 15623	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ⊆ wss 3944 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 +∞cpnf 11267 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 3c3 12290 ℤcz 12580 ℝ⁺crp 12998 [,)cico 13350 ...cfz 13508 ⌊cfl 13779 seqcseq 13990 abscabs 15205 ⇝ cli 15452 𝑂(1)co1 15454 Σcsu 15656 Basecbs 17171 0_gc0g 17412 ℤRHomczrh 21412 ℤ/nℤczn 21415 logclog 26475 Λcvma 27011 μcmu 27014 DChrcdchr 27152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-dju 9916 df-card 9954 df-acn 9957 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-shft 15038 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-o1 15458 df-lo1 15459 df-sum 15657 df-ef 16035 df-e 16036 df-sin 16037 df-cos 16038 df-tan 16039 df-pi 16040 df-dvds 16223 df-gcd 16461 df-prm 16634 df-pc 16797 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-qus 17482 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-nsg 19070 df-eqg 19071 df-ghm 19159 df-cntz 19259 df-od 19474 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-2idl 21133 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-fbas 21263 df-fg 21264 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-zn 21419 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-perf 23028 df-cn 23118 df-cnp 23119 df-haus 23206 df-cmp 23278 df-tx 23453 df-hmeo 23646 df-fil 23737 df-fm 23829 df-flim 23830 df-flf 23831 df-xms 24213 df-ms 24214 df-tms 24215 df-cncf 24785 df-limc 25782 df-dv 25783 df-ulm 26300 df-log 26477 df-cxp 26478 df-atan 26786 df-em 26912 df-vma 27017 df-mu 27020 df-dchr 27153

This theorem is referenced by: dchrvmasumif 27423

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