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Theorem dchrisum0lem2 26102

Description: Lemma for dchrisum0 26104. (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 11703	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
2		rpcn 12387	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
3	2	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
4		fzfid 13336	. . . . 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 4008	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
11		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
12	10, 11	sseldi 3913	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
13	12	eldifad 3893	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
14	13	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
15		elfzelz 12902	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
16	15	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℤ)
17	5, 6, 7, 8, 14, 16	dchrzrhcl 25829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
18		elfznn 12931	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
19	18	nnrpd 12417	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
20	19	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℝ⁺)
21	20	rpcnd 12421	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℂ)
22	20	rpne0d 12424	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ≠ 0)
23	17, 21, 22	divcld 11405	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
24	4, 23	fsumcl 15082	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
25	3, 24	mulcld 10650	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ)
26		rpssre 12384	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
27		2cn 11700	. . . . 5 ⊢ 2 ∈ ℂ
28		o1const 14968	. . . . 5 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
29	26, 27, 28	mp2an 691	. . . 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 10631	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
33		dchrisum0lem2.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
34		elrege0 12832	. . . . . 6 ⊢ (𝐸 ∈ (0[,)+∞) ↔ (𝐸 ∈ ℝ ∧ 0 ≤ 𝐸))
35	34	simplbi 501	. . . . 5 ⊢ (𝐸 ∈ (0[,)+∞) → 𝐸 ∈ ℝ)
36	33, 35	syl 17	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ)
37	3, 24	absmuld 14806	. . . . . . 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 12392	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
39	38	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
40		absid 14648	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
41	39, 40	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘𝑥) = 𝑥)
42	41	oveq1d 7150	. . . . . . 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 2833	. . . . . 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 716	. . . . 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 716	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
46	45	subid1d 10975	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))
47	18	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
48		2fveq3 6650	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
49		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
50	48, 49	oveq12d 7153	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
51		dchrisum0lem2.k	. . . . . . . . . . . . . 14 ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
52		ovex 7168	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘(𝐿‘𝑎)) / 𝑎) ∈ V
53	50, 51, 52	fvmpt3i 6750	. . . . . . . . . . . . 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 720	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
56		rpregt0 12391	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
57	56	ad2antrl 727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
58	57	simpld 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
59		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
60		flge1nn 13186	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
61	58, 59, 60	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ ℕ)
62		nnuz 12269	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
63	61, 62	eleqtrdi 2900	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
64	23	adantlrr 720	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
65	55, 63, 64	fsumser 15079	. . . . . . . . . 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 4683	. . . . . . . . . . . . . . 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 26088	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑇 = 0))
73	11, 72	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 = 0)
74	73	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑇 = 0)
75	74	eqcomd 2804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 0 = 𝑇)
76	65, 75	oveq12d 7153	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
77	46, 76	eqtr3d 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))
78	77	fveq2d 6649	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
79		2fveq3 6650	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑥)))
80	79	fvoveq1d 7157	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)))
81		oveq2 7143	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝐸 / 𝑦) = (𝐸 / 𝑥))
82	80, 81	breq12d 5043	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥)))
83	71	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦))
84		1re 10630	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
85		elicopnf 12823	. . . . . . . . . 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 586	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ (1[,)+∞))
88	82, 83, 87	rspcdva 3573	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥))
89	78, 88	eqbrtrd 5052	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥))
90	45	abscld 14788	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ)
91	36	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝐸 ∈ ℝ)
92		lemuldiv2 11510	. . . . . . 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 1368	. . . . . 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 5052	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸)
96	31, 25, 32, 36, 95	elo1d 14885	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ 𝑂(1))
97	1, 25, 30, 96	o1mul2 14973	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1))
98		fzfid 13336	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
99	20	rpsqrtcld 14763	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
100	99	rpcnd 12421	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
101	99	rpne0d 12424	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
102	17, 100, 101	divcld 11405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
103	102	adantr 484	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
104		elfznn 12931	. . . . . . . . . 10 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
105	104	adantl 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ)
106	105	nnrpd 12417	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
107	106	rpsqrtcld 14763	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
108	107	rpcnd 12421	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℂ)
109	107	rpne0d 12424	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ≠ 0)
110	103, 108, 109	divcld 11405	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
111	98, 110	fsumcl 15082	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
112	4, 111	fsumcl 15082	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
113		mulcl 10610	. . . 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 590	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ)
115		2re 11699	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
116		simpr 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
117		2z 12002	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℤ
118		rpexpcl 13444	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
119	116, 117, 118	sylancl 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
120		rpdivcl 12402	. . . . . . . . . . . . 13 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
121	119, 19, 120	syl2an 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
122	121	rpsqrtcld 14763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ⁺)
123	122	rpred 12419	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) ∈ ℝ)
124		remulcl 10611	. . . . . . . . . 10 ⊢ ((2 ∈ ℝ ∧ (√‘((𝑥↑2) / 𝑚)) ∈ ℝ) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
125	115, 123, 124	sylancr 590	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ)
126	125	recnd 10658	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℂ)
127	102, 126	mulcld 10650	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))) ∈ ℂ)
128	4, 111, 127	fsumsub 15135	. . . . . 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 12428	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
130		reccl 11294	. . . . . . . . . . 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 15082	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) ∈ ℂ)
133	102, 132, 126	subdid 11085	. . . . . . . 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 6645	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (⌊‘𝑦) = (⌊‘((𝑥↑2) / 𝑚)))
135	134	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (1...(⌊‘𝑦)) = (1...(⌊‘((𝑥↑2) / 𝑚))))
136	135	sumeq1d 15050	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)))
137		fveq2 6645	. . . . . . . . . . . . 13 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑚)))
138	137	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (2 · (√‘𝑦)) = (2 · (√‘((𝑥↑2) / 𝑚))))
139	136, 138	oveq12d 7153	. . . . . . . . . . 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 7168	. . . . . . . . . . 11 ⊢ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))) ∈ V
142	139, 140, 141	fvmpt3i 6750	. . . . . . . . . 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 7151	. . . . . . . 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 11408	. . . . . . . . . . 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 15052	. . . . . . . . . 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 15131	. . . . . . . . . 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 2836	. . . . . . . . 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 7150	. . . . . . . 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 2843	. . . . . . 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 15052	. . . . . 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 10610	. . . . . . . . . 10 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
153	27, 3, 152	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝑥) ∈ ℂ)
154	4, 153, 23	fsummulc2 15131	. . . . . . . 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 10653	. . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · 𝑥) ∈ ℂ)
157	156, 102, 100, 101	div12d 11441	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚))))
158	99	rpcnne0d 12428	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
159		divdiv1 11340	. . . . . . . . . . . . 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 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))))
161	20	rprege0d 12426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
162		remsqsqrt 14608	. . . . . . . . . . . . . 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 7151	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
165	160, 164	eqtr2d 2834	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)))
166	165	oveq2d 7151	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))))
167	119	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥↑2) ∈ ℝ⁺)
168	167	rprege0d 12426	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
169		sqrtdiv 14617	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
170	168, 20, 169	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
171	38	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
172		sqrtsq 14621	. . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
175	170, 174	eqtrd 2833	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
176	175	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = (2 · (𝑥 / (√‘𝑚))))
177		2cnd 11703	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
178	3	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
179		divass 11305	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
180	177, 178, 158, 179	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚))))
181	176, 180	eqtr4d 2836	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (2 · (√‘((𝑥↑2) / 𝑚))) = ((2 · 𝑥) / (√‘𝑚)))
182	181	oveq2d 7151	. . . . . . . . . 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 2843	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 · (√‘((𝑥↑2) / 𝑚)))))
184	183	sumeq2dv 15052	. . . . . . . 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 2841	. . . . . . 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 7151	. . . . . 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 2843	. . . . 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 5125	. . . 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 26101	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))
195	188, 194	eqeltrrd 2891	. . 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 14978	. 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

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {crab 3110 ∖ cdif 3878 ⊆ wss 3881 {csn 4525 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 +∞cpnf 10661 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 [,)cico 12728 ...cfz 12885 ⌊cfl 13155 seqcseq 13364 ↑cexp 13425 √csqrt 14584 abscabs 14585 ⇝ cli 14833 ⇝_𝑟 crli 14834 𝑂(1)co1 14835 Σcsu 15034 Basecbs 16475 0_gc0g 16705 ℤRHomczrh 20193 ℤ/nℤczn 20196 DChrcdchr 25816

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-o1 14839 df-lo1 14840 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-qus 16774 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-cntz 18439 df-od 18648 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zn 20200 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149 df-dchr 25817

This theorem is referenced by: dchrisum0lem3 26103

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