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Theorem dchrisum0lem2 27875

Description: Lemma for dchrisum0 27877. (Contributed by Mario Carneiro, 12-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum2.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum2.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum2.1	⊢ 1 = (0_g‘𝐺)
rpvmasum2.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
dchrisum0.b	⊢ (𝜑 → 𝑋 ∈ 𝑊)
dchrisum0lem1.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
dchrisum0.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrisum0.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrisum0.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
dchrisum0lem2.h	⊢ 𝐻 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
dchrisum0lem2.u	⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
dchrisum0lem2.k	⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrisum0lem2.e	⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
dchrisum0lem2.t	⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
dchrisum0lem2.3	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))

**Assertion**
Ref	Expression
dchrisum0lem2	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

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

**Proof of Theorem dchrisum0lem2**
Step	Hyp	Ref	Expression
1		2cnd 12448	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
2		rpcn 13158	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
3	2	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
4		fzfid 14143	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
5		rpvmasum2.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
6		rpvmasum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
7		rpvmasum2.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
8		rpvmasum.l	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑍)
9		rpvmasum2.w	. . . . . . . . . . 11 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
10	9	ssrab3 4010	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
11		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
12	10, 11	sselid 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
13	12	eldifad 4054	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	13	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
15		elfzelz 13683	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
16	15	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℤ)
17	5, 6, 7, 8, 14, 16	dchrzrhcl 27602	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
18		elfznn 13714	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
19	18	nnrpd 13189	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
20	19	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	rpcnd 13193	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℂ)
22	20	rpne0d 13196	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ≠ 0)
23	17, 21, 22	divcld 12120	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
24	4, 23	fsumcl 15928	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
25	3, 24	mulcld 11356	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
26		rpssre 13155	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
27		2cn 12445	. . . . 5 ⊢ 2 ∈ ℂ
28		o1const 15816	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
29	26, 27, 28	mp2an 705	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
30	29	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
31	26	a1i 11	. . . 4 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
32		1red 11336	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
33		dchrisum0lem2.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
34		elrege0 13612	. . . . . 6 ⊢ (𝐸 ∈ (0[,)+∞) ↔ (𝐸 ∈ ℝ ∧ 0 ≤ 𝐸))
35	34	simplbi 502	. . . . 5 ⊢ (𝐸 ∈ (0[,)+∞) → 𝐸 ∈ ℝ)
36	33, 35	syl 18	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
37	3, 24	absmuld 15653	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = ((abs‘𝑥) · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))
38		rprege0 13163	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
39	38	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
40		absid 15492	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
41	39, 40	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘𝑥) = 𝑥)
42	41	oveq1d 7458	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((abs‘𝑥) · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))
43	37, 42	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))
44	43	adantrr 730	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))
45	24	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
46	45	subid1d 11685	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))
47	18	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
48		2fveq3 6913	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
49		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
50	48, 49	oveq12d 7461	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
51		dchrisum0lem2.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
52		ovex 7476	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘(𝐿‘𝑎)) / 𝑎) ∈ V
53	50, 51, 52	fvmpt3i 7022	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
54	47, 53	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
55	54	adantlrr 734	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
56		rpregt0 13162	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
57	56	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
58	57	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
59		simprr 785	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
60		flge1nn 13988	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
61	58, 59, 60	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
62		nnuz 13031	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
63	61, 62	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
64	23	adantlrr 734	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
65	55, 63, 64	fsumser 15925	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐾)‘(⌊‘𝑥)))
66		rpvmasum.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
67		rpvmasum2.1	. . . . . . . . . . . . . 14 ⊢ 1 = (0_g‘𝐺)
68		eldifsni 4753	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
69	12, 68	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ≠ 1 )
70		dchrisum0lem2.t	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
71		dchrisum0lem2.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))
72	6, 8, 66, 5, 7, 67, 13, 69, 51, 33, 70, 71, 9	dchrvmaeq0 27861	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑇 = 0))
73	11, 72	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = 0)
74	73	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 = 0)
75	74	eqcomd 2769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 = 𝑇)
76	65, 75	oveq12d 7461	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
77	46, 76	eqtr3d 2800	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
78	77	fveq2d 6912	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
79		2fveq3 6913	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑥)))
80	79	fvoveq1d 7465	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
81		oveq2 7451	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐸 / 𝑦) = (𝐸 / 𝑥))
82	80, 81	breq12d 5119	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥)))
83	71	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))
84		1re 11335	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
85		elicopnf 13603	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
86	84, 85	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
87	58, 59, 86	sylanbrc 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ (1[,)+∞))
88	82, 83, 87	rspcdva 3581	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥))
89	78, 88	eqbrtrd 5130	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥))
90	45	abscld 15635	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ)
91	36	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐸 ∈ ℝ)
92		lemuldiv2 12225	. . . . . . 7 ⊢ (((abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ ∧ 𝐸 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → ((𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸 ↔ (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥)))
93	90, 91, 57, 92	syl3anc 1399	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸 ↔ (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥)))
94	89, 93	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸)
95	44, 94	eqbrtrd 5130	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸)
96	31, 25, 32, 36, 95	elo1d 15732	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ 𝑂(1))
97	1, 25, 30, 96	o1mul2 15821	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1))
98		fzfid 14143	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
99	20	rpsqrtcld 15608	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
100	99	rpcnd 13193	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
101	99	rpne0d 13196	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
102	17, 100, 101	divcld 12120	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
103	102	adantr 486	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
104		elfznn 13714	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
105	104	adantl 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ)
106	105	nnrpd 13189	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
107	106	rpsqrtcld 15608	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
108	107	rpcnd 13193	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℂ)
109	107	rpne0d 13196	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ≠ 0)
110	103, 108, 109	divcld 12120	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
111	98, 110	fsumcl 15928	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
112	4, 111	fsumcl 15928	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
113		mulcl 11311	. . . 4 ⊢ ((2 ∈ ℂ ∧ (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) → (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ)
114	27, 25, 113	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ)
115		2re 12444	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
116		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
117		2z 12755	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
118		rpexpcl 14250	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
119	116, 117, 118	sylancl 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
120		rpdivcl 13174	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
121	119, 19, 120	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
122	121	rpsqrtcld 15608	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ⁺)
123	122	rpred 13191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ)
124		remulcl 11312	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (√‘((𝑥↑2) / 𝑚)) ∈ ℝ) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
125	115, 123, 124	sylancr 599	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
126	125	recnd 11364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℂ)
127	102, 126	mulcld 11356	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))) ∈ ℂ)
128	4, 111, 127	fsumsub 15983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
129	107	rpcnne0d 13200	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
130		reccl 12008	. . . . . . . . . . 11 ⊢ (((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0) → (1 / (√‘𝑑)) ∈ ℂ)
131	129, 130	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (1 / (√‘𝑑)) ∈ ℂ)
132	98, 131	fsumcl 15928	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) ∈ ℂ)
133	102, 132, 126	subdid 11799	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))) = ((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
134		fveq2 6908	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (⌊‘𝑦) = (⌊‘((𝑥↑2) / 𝑚)))
135	134	oveq2d 7459	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (1...(⌊‘𝑦)) = (1...(⌊‘((𝑥↑2) / 𝑚))))
136	135	sumeq1d 15896	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)))
137		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑚)))
138	137	oveq2d 7459	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (2 · (√‘𝑦)) = (2 · (√‘((𝑥↑2) / 𝑚))))
139	136, 138	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚)))))
140		dchrisum0lem2.h	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
141		ovex 7476	. . . . . . . . . . 11 ⊢ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))) ∈ V
142	139, 140, 141	fvmpt3i 7022	. . . . . . . . . 10 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ⁺ → (𝐻‘((𝑥↑2) / 𝑚)) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚)))))
143	121, 142	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚)))))
144	143	oveq2d 7459	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))))
145	103, 108, 109	divrecd 12123	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑))))
146	145	sumeq2dv 15898	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑))))
147	98, 102, 131	fsummulc2 15979	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑))))
148	146, 147	eqtr4d 2801	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))))
149	148	oveq1d 7458	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))) = ((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
150	133, 144, 149	3eqtr4d 2808	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
151	150	sumeq2dv 15898	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
152		mulcl 11311	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
153	27, 3, 152	sylancr 599	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝑥) ∈ ℂ)
154	4, 153, 23	fsummulc2 15979	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝑥) · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))
155	1, 3, 24	mulassd 11359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝑥) · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))
156	153	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · 𝑥) ∈ ℂ)
157	156, 102, 100, 101	div12d 12156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚))))
158	99	rpcnne0d 13200	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
159		divdiv1 12055	. . . . . . . . . . . . 13 ⊢ (((𝑋‘(𝐿‘𝑚)) ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))))
160	17, 158, 158, 159	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))))
161	20	rprege0d 13198	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
162		remsqsqrt 15452	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → ((√‘𝑚) · (√‘𝑚)) = 𝑚)
163	161, 162	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) · (√‘𝑚)) = 𝑚)
164	163	oveq2d 7459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
165	160, 164	eqtr2d 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)))
166	165	oveq2d 7459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))))
167	119	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥↑2) ∈ ℝ⁺)
168	167	rprege0d 13198	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
169		sqrtdiv 15461	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
170	168, 20, 169	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
171	38	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
172		sqrtsq 15465	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘(𝑥↑2)) = 𝑥)
173	171, 172	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘(𝑥↑2)) = 𝑥)
174	173	oveq1d 7458	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
175	170, 174	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
176	175	oveq2d 7459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = (2 · (𝑥 / (√‘𝑚))))
177		2cnd 12448	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
178	3	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
179		divass 12019	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
180	177, 178, 158, 179	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
181	176, 180	eqtr4d 2801	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = ((2 · 𝑥) / (√‘𝑚)))
182	181	oveq2d 7459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚))))
183	157, 166, 182	3eqtr4d 2808	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))))
184	183	sumeq2dv 15898	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))))
185	154, 155, 184	3eqtr3d 2806	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))))
186	185	oveq2d 7459	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚))))))
187	128, 151, 186	3eqtr4d 2808	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))))
188	187	mpteq2dva 5201	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))))
189		dchrisum0lem1.f	. . . . 5 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
190		dchrisum0.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
191		dchrisum0.s	. . . . 5 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
192		dchrisum0.1	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
193		dchrisum0lem2.u	. . . . 5 ⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
194	6, 8, 66, 5, 7, 67, 9, 11, 189, 190, 191, 192, 140, 193	dchrisum0lem2a 27874	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))
195	188, 194	eqeltrrd 2865	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))) ∈ 𝑂(1))
196	112, 114, 195	o1dif 15826	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1)))
197	97, 196	mpbird 260	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 {crab 3416 ⊆ wss 3900 ∖ cdif 4048 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 ℝcr 11226 0cc0 11227 1c1 11228 + caddc 11230 · cmul 11232 +∞cpnf 11367 < clt 11370 ≤ cle 11371 − cmin 11568 / cdiv 12000 ℕcn 12362 2c2 12424 ℤcz 12720 ℤ_≥cuz 12992 ℝ⁺crp 13147 [,)cico 13505 ...cfz 13666 ⌊cfl 13957 seqcseq 14171 ↑cexp 14231 √csqrt 15429 abscabs 15430 ⇝ cli 15680 ⇝_𝑟 crli 15681 𝑂(1)co1 15682 Σcsu 15882 Basecbs 17416 0_gc0g 17639 ℤRHomczrh 21835 ℤ/nℤczn 21838 DChrcdchr 27589

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305 ax-addf 11306 ax-mulf 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-disj 5074 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8745 df-ec 8747 df-qs 8751 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-fi 9430 df-sup 9461 df-inf 9462 df-oi 9531 df-card 10047 df-acn 10050 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-q 13103 df-rp 13148 df-xneg 13268 df-xadd 13269 df-xmul 13270 df-ioo 13507 df-ioc 13508 df-ico 13509 df-icc 13510 df-fz 13667 df-fzo 13816 df-fl 13959 df-mod 14037 df-seq 14172 df-exp 14232 df-fac 14445 df-bc 14474 df-hash 14502 df-shft 15249 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-rlim 15685 df-o1 15686 df-lo1 15687 df-sum 15883 df-ef 16262 df-sin 16264 df-cos 16265 df-pi 16267 df-dvds 16452 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-hom 17481 df-cco 17482 df-rest 17622 df-topn 17623 df-0g 17641 df-gsum 17642 df-topgen 17643 df-pt 17644 df-prds 17647 df-xrs 17703 df-qtop 17708 df-imas 17709 df-qus 17710 df-xps 17711 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-nsg 19363 df-eqg 19364 df-ghm 19457 df-cntz 19561 df-od 19772 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-rhm 20732 df-subrng 20828 df-subrg 20852 df-drng 21012 df-lmod 21167 df-lss 21237 df-lsp 21277 df-sra 21478 df-rgmod 21479 df-lidl 21516 df-rsp 21517 df-2idl 21573 df-psmet 21700 df-xmet 21701 df-met 21702 df-bl 21703 df-mopn 21704 df-fbas 21705 df-fg 21706 df-cnfld 21709 df-zring 21783 df-zrh 21839 df-zn 21842 df-top 23242 df-topon 23259 df-topsp 23281 df-bases 23294 df-cld 23367 df-ntr 23368 df-cls 23369 df-nei 23446 df-lp 23484 df-perf 23485 df-cn 23575 df-cnp 23576 df-haus 23663 df-cmp 23735 df-tx 23911 df-hmeo 24104 df-fil 24195 df-fm 24287 df-flim 24288 df-flf 24289 df-xms 24669 df-ms 24670 df-tms 24671 df-cncf 25229 df-limc 26216 df-dv 26217 df-log 26914 df-cxp 26915 df-dchr 27590

This theorem is used by: dchrisum0lem3 27876

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