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Theorem dchrisum0lem2 27469

Description: Lemma for dchrisum0 27471. (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 12248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
2		rpcn 12942	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
3	2	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
4		fzfid 13924	. . . . 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 4015	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
11		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
12	10, 11	sselid 3915	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
13	12	eldifad 3897	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	13	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
15		elfzelz 13467	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
16	15	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℤ)
17	5, 6, 7, 8, 14, 16	dchrzrhcl 27196	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
18		elfznn 13496	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
19	18	nnrpd 12973	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
20	19	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	rpcnd 12977	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℂ)
22	20	rpne0d 12980	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ≠ 0)
23	17, 21, 22	divcld 11920	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
24	4, 23	fsumcl 15684	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
25	3, 24	mulcld 11154	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
26		rpssre 12939	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
27		2cn 12245	. . . . 5 ⊢ 2 ∈ ℂ
28		o1const 15571	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
29	26, 27, 28	mp2an 693	. . . 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 11134	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
33		dchrisum0lem2.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
34		elrege0 13396	. . . . . 6 ⊢ (𝐸 ∈ (0[,)+∞) ↔ (𝐸 ∈ ℝ ∧ 0 ≤ 𝐸))
35	34	simplbi 496	. . . . 5 ⊢ (𝐸 ∈ (0[,)+∞) → 𝐸 ∈ ℝ)
36	33, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
37	3, 24	absmuld 15408	. . . . . . 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 12947	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
39	38	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
40		absid 15247	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
41	39, 40	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘𝑥) = 𝑥)
42	41	oveq1d 7371	. . . . . . 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 2770	. . . . . 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 718	. . . . 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 718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
46	45	subid1d 11483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))
47	18	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
48		2fveq3 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
49		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
50	48, 49	oveq12d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
51		dchrisum0lem2.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
52		ovex 7389	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘(𝐿‘𝑎)) / 𝑎) ∈ V
53	50, 51, 52	fvmpt3i 6942	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
54	47, 53	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
55	54	adantlrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
56		rpregt0 12946	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
57	56	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
58	57	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
59		simprr 773	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
60		flge1nn 13769	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
61	58, 59, 60	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
62		nnuz 12816	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
63	61, 62	eleqtrdi 2845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
64	23	adantlrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
65	55, 63, 64	fsumser 15681	. . . . . . . . . 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 4725	. . . . . . . . . . . . . . 15 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
69	12, 68	syl 17	. . . . . . . . . . . . . 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 27455	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑇 = 0))
73	11, 72	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = 0)
74	73	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 = 0)
75	74	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 = 𝑇)
76	65, 75	oveq12d 7374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
77	46, 76	eqtr3d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
78	77	fveq2d 6833	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
79		2fveq3 6834	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑥)))
80	79	fvoveq1d 7378	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
81		oveq2 7364	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐸 / 𝑦) = (𝐸 / 𝑥))
82	80, 81	breq12d 5087	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥)))
83	71	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))
84		1re 11133	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
85		elicopnf 13387	. . . . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ (1[,)+∞))
88	82, 83, 87	rspcdva 3563	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥))
89	78, 88	eqbrtrd 5096	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥))
90	45	abscld 15390	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ)
91	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐸 ∈ ℝ)
92		lemuldiv2 12026	. . . . . . 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 1374	. . . . . 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 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸)
95	44, 94	eqbrtrd 5096	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸)
96	31, 25, 32, 36, 95	elo1d 15487	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ 𝑂(1))
97	1, 25, 30, 96	o1mul2 15576	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1))
98		fzfid 13924	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
99	20	rpsqrtcld 15363	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
100	99	rpcnd 12977	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
101	99	rpne0d 12980	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
102	17, 100, 101	divcld 11920	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
103	102	adantr 480	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
104		elfznn 13496	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
105	104	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ)
106	105	nnrpd 12973	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
107	106	rpsqrtcld 15363	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
108	107	rpcnd 12977	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℂ)
109	107	rpne0d 12980	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ≠ 0)
110	103, 108, 109	divcld 11920	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
111	98, 110	fsumcl 15684	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
112	4, 111	fsumcl 15684	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
113		mulcl 11111	. . . 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 588	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ)
115		2re 12244	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
116		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
117		2z 12548	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
118		rpexpcl 14031	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
119	116, 117, 118	sylancl 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
120		rpdivcl 12958	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
121	119, 19, 120	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
122	121	rpsqrtcld 15363	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ⁺)
123	122	rpred 12975	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ)
124		remulcl 11112	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (√‘((𝑥↑2) / 𝑚)) ∈ ℝ) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
125	115, 123, 124	sylancr 588	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
126	125	recnd 11162	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℂ)
127	102, 126	mulcld 11154	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))) ∈ ℂ)
128	4, 111, 127	fsumsub 15739	. . . . . 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 12984	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
130		reccl 11805	. . . . . . . . . . 11 ⊢ (((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0) → (1 / (√‘𝑑)) ∈ ℂ)
131	129, 130	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (1 / (√‘𝑑)) ∈ ℂ)
132	98, 131	fsumcl 15684	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) ∈ ℂ)
133	102, 132, 126	subdid 11595	. . . . . . . 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 6829	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (⌊‘𝑦) = (⌊‘((𝑥↑2) / 𝑚)))
135	134	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (1...(⌊‘𝑦)) = (1...(⌊‘((𝑥↑2) / 𝑚))))
136	135	sumeq1d 15651	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)))
137		fveq2 6829	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑚)))
138	137	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (2 · (√‘𝑦)) = (2 · (√‘((𝑥↑2) / 𝑚))))
139	136, 138	oveq12d 7374	. . . . . . . . . . 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 7389	. . . . . . . . . . 11 ⊢ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))) ∈ V
142	139, 140, 141	fvmpt3i 6942	. . . . . . . . . 10 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ⁺ → (𝐻‘((𝑥↑2) / 𝑚)) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚)))))
143	121, 142	syl 17	. . . . . . . . 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 7372	. . . . . . . 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 11923	. . . . . . . . . . 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 15653	. . . . . . . . . 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 15735	. . . . . . . . . 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 2773	. . . . . . . . 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 7371	. . . . . . . 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 2780	. . . . . . 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 15653	. . . . . 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 11111	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
153	27, 3, 152	sylancr 588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝑥) ∈ ℂ)
154	4, 153, 23	fsummulc2 15735	. . . . . . . 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 11157	. . . . . . . 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 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · 𝑥) ∈ ℂ)
157	156, 102, 100, 101	div12d 11956	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚))))
158	99	rpcnne0d 12984	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
159		divdiv1 11855	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))))
161	20	rprege0d 12982	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
162		remsqsqrt 15207	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) → ((√‘𝑚) · (√‘𝑚)) = 𝑚)
163	161, 162	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) · (√‘𝑚)) = 𝑚)
164	163	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
165	160, 164	eqtr2d 2771	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)))
166	165	oveq2d 7372	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))))
167	119	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥↑2) ∈ ℝ⁺)
168	167	rprege0d 12982	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
169		sqrtdiv 15216	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
170	168, 20, 169	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
171	38	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
172		sqrtsq 15220	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘(𝑥↑2)) = 𝑥)
173	171, 172	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘(𝑥↑2)) = 𝑥)
174	173	oveq1d 7371	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
175	170, 174	eqtrd 2770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
176	175	oveq2d 7372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = (2 · (𝑥 / (√‘𝑚))))
177		2cnd 12248	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
178	3	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
179		divass 11816	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
180	177, 178, 158, 179	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
181	176, 180	eqtr4d 2773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = ((2 · 𝑥) / (√‘𝑚)))
182	181	oveq2d 7372	. . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))))
184	183	sumeq2dv 15653	. . . . . . . 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 2778	. . . . . . 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 7372	. . . . . 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 2780	. . . . 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 5167	. . . 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 27468	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))
195	188, 194	eqeltrrd 2836	. . 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 15581	. 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 257	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2930 ∀wral 3049 {crab 3387 ∖ cdif 3882 ⊆ wss 3885 {csn 4557 class class class wbr 5074 ↦ cmpt 5155 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 +∞cpnf 11165 < clt 11168 ≤ cle 11169 − cmin 11366 / cdiv 11796 ℕcn 12163 2c2 12225 ℤcz 12513 ℤ_≥cuz 12777 ℝ⁺crp 12931 [,)cico 13289 ...cfz 13450 ⌊cfl 13738 seqcseq 13952 ↑cexp 14012 √csqrt 15184 abscabs 15185 ⇝ cli 15435 ⇝_𝑟 crli 15436 𝑂(1)co1 15437 Σcsu 15637 Basecbs 17168 0_gc0g 17391 ℤRHomczrh 21468 ℤ/nℤczn 21471 DChrcdchr 27183

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8632 df-ec 8634 df-qs 8638 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-o1 15441 df-lo1 15442 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-dvds 16211 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-qus 17462 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-nsg 19089 df-eqg 19090 df-ghm 19177 df-cntz 19281 df-od 19492 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-oppr 20306 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-rhm 20441 df-subrng 20512 df-subrg 20536 df-drng 20697 df-lmod 20846 df-lss 20916 df-lsp 20956 df-sra 21157 df-rgmod 21158 df-lidl 21195 df-rsp 21196 df-2idl 21237 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-zring 21416 df-zrh 21472 df-zn 21475 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-cmp 23340 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-cxp 26509 df-dchr 27184

This theorem is referenced by: dchrisum0lem3 27470

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