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Theorem dchrisum0lem1b 25204

Description: Lemma for dchrisum0lem1 25205. (Contributed by Mario Carneiro, 7-Jun-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)))

**Assertion**
Ref	Expression
dchrisum0lem1b	⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))

Distinct variable groups: 𝑥,𝑚,𝑦, 1 𝑚,𝑑,𝑥,𝑦,𝐶 𝐹,𝑑,𝑥,𝑦 𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝑁,𝑥,𝑦 𝜑,𝑑,𝑚,𝑥 𝑆,𝑑,𝑚,𝑥,𝑦 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) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑚,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑊(𝑦,𝑚,𝑎,𝑑) 𝑍(𝑎,𝑑)

**Proof of Theorem dchrisum0lem1b**
Step	Hyp	Ref	Expression
1		fzfid 12772	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
2		ssun2 3777	. . . . . . 7 ⊢ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
3		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
4	3	rprege0d 11879	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5		flge0nn0 12621	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
7		nn0p1nn 11332	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
9	8	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ ℕ)
10		nnuz 11723	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
11	9, 10	syl6eleq 2711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
12		dchrisum0lem1a 25175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))))
13	12	simprd 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥)))
14		fzsplit2 12366	. . . . . . . 8 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
15	11, 13, 14	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
16	2, 15	syl5sseqr 3654	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
17	16	sselda 3603	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
18		rpvmasum2.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
19		rpvmasum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20		rpvmasum2.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
21		rpvmasum.l	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑍)
22		rpvmasum2.w	. . . . . . . . . . 11 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
23		ssrab2 3687	. . . . . . . . . . 11 ⊢ {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} ⊆ (𝐷 ∖ { 1 })
24	22, 23	eqsstri 3635	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
25		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
26	24, 25	sseldi 3601	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
27	26	eldifad 3586	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
28	27	ad3antrrr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑋 ∈ 𝐷)
29		elfzelz 12342	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℤ)
30	29	adantl 482	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℤ)
31	18, 19, 20, 21, 28, 30	dchrzrhcl 24970	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
32		elfznn 12370	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℕ)
33	32	adantl 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℕ)
34	33	nnrpd 11870	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℝ⁺)
35	34	rpsqrtcld 14150	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℝ⁺)
36	35	rpcnd 11874	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℂ)
37	35	rpne0d 11877	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ≠ 0)
38	31, 36, 37	divcld 10801	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
39	17, 38	syldan 487	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
40	1, 39	fsumcl 14464	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
41	40	abscld 14175	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
42		1zzd 11408	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
43	27	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
44		nnz 11399	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
45	44	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
46	18, 19, 20, 21, 43, 45	dchrzrhcl 24970	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
47		nnrp 11842	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
48	47	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
49	48	rpsqrtcld 14150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
50	49	rpcnd 11874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
51	49	rpne0d 11877	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
52	46, 50, 51	divcld 10801	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
53		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
54		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑚 → (𝐿‘𝑎) = (𝐿‘𝑚))
55	54	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
56		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑚 → (√‘𝑎) = (√‘𝑚))
57	55, 56	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
58	57	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
59	53, 58	eqtri 2644	. . . . . . . . . 10 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
60	52, 59	fmptd 6385	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
61	60	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
62	10, 42, 61	serf 12829	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
63	62	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → seq1( + , 𝐹):ℕ⟶ℂ)
64	3	rpregt0d 11878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
65	64	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
66	65	simpld 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
67		1red 10055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
68		elfznn 12370	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
69	68	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
70	69	nnred 11035	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ)
71	69	nnge1d 11063	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑑)
72	3	rpred 11872	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
73		fznnfl 12661	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
74	72, 73	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
75	74	simplbda 654	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
76	67, 70, 66, 71, 75	letrd 10194	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑥)
77		flge1nn 12622	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
78	66, 76, 77	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ ℕ)
79		eluznn 11758	. . . . . . 7 ⊢ (((⌊‘𝑥) ∈ ℕ ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℕ)
80	78, 13, 79	syl2anc 693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℕ)
81	63, 80	ffvelrnd 6360	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) ∈ ℂ)
82		dchrisum0.s	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
83		climcl 14230	. . . . . . 7 ⊢ (seq1( + , 𝐹) ⇝ 𝑆 → 𝑆 ∈ ℂ)
84	82, 83	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
85	84	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑆 ∈ ℂ)
86	81, 85	subcld 10392	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆) ∈ ℂ)
87	86	abscld 14175	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ∈ ℝ)
88	63, 78	ffvelrnd 6360	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘𝑥)) ∈ ℂ)
89	85, 88	subcld 10392	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ ℂ)
90	89	abscld 14175	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) ∈ ℝ)
91	87, 90	readdcld 10069	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ∈ ℝ)
92		2re 11090	. . . . . 6 ⊢ 2 ∈ ℝ
93		dchrisum0.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
94		elrege0 12278	. . . . . . . 8 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
95	93, 94	sylib 208	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
96	95	simpld 475	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
97		remulcl 10021	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (2 · 𝐶) ∈ ℝ)
98	92, 96, 97	sylancr 695	. . . . 5 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
99	98	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℝ)
100	3	rpsqrtcld 14150	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
101	99, 100	rerpdivcld 11903	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
102	101	adantr 481	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
103		ssun1 3776	. . . . . . . . . . 11 ⊢ (1...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
104	103, 15	syl5sseqr 3654	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
105	104	sselda 3603	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
106		ovex 6678	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) ∈ V
107	57, 53, 106	fvmpt3i 6287	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
108	33, 107	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
109	105, 108	syldan 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
110	78, 10	syl6eleq 2711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
111	105, 38	syldan 487	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
112	109, 110, 111	fsumser 14461	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
113	112, 88	eqeltrd 2701	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
114	113, 40	pncan2d 10394	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
115		reflcl 12597	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
116	66, 115	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ ℝ)
117	116	ltp1d 10954	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) < ((⌊‘𝑥) + 1))
118		fzdisj 12368	. . . . . . . . 9 ⊢ ((⌊‘𝑥) < ((⌊‘𝑥) + 1) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) = ∅)
119	117, 118	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) = ∅)
120		fzfid 12772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
121	119, 15, 120, 38	fsumsplit 14471	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
122	80, 10	syl6eleq 2711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘1))
123	108, 122, 38	fsumser 14461	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))))
124	121, 123	eqtr3d 2658	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))))
125	124, 112	oveq12d 6668	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − (seq1( + , 𝐹)‘(⌊‘𝑥))))
126	114, 125	eqtr3d 2658	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = ((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − (seq1( + , 𝐹)‘(⌊‘𝑥))))
127	126	fveq2d 6195	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − (seq1( + , 𝐹)‘(⌊‘𝑥)))))
128	81, 88, 85	abs3difd 14199	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − (seq1( + , 𝐹)‘(⌊‘𝑥)))) ≤ ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))))
129	127, 128	eqbrtrd 4675	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))))
130	96	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
131		simplr 792	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
132	131	rpsqrtcld 14150	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑥) ∈ ℝ⁺)
133	130, 132	rerpdivcld 11903	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘𝑥)) ∈ ℝ)
134		2z 11409	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
135		rpexpcl 12879	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
136	3, 134, 135	sylancl 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
137	136	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥↑2) ∈ ℝ⁺)
138	69	nnrpd 11870	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
139	137, 138	rpdivcld 11889	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ ℝ⁺)
140	139	rpsqrtcld 14150	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑑)) ∈ ℝ⁺)
141	130, 140	rerpdivcld 11903	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘((𝑥↑2) / 𝑑))) ∈ ℝ)
142	136	rpred 11872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ)
143		nndivre 11056	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((𝑥↑2) / 𝑑) ∈ ℝ)
144	142, 68, 143	syl2an 494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ ℝ)
145	12	simpld 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ≤ ((𝑥↑2) / 𝑑))
146	67, 66, 144, 76, 145	letrd 10194	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ ((𝑥↑2) / 𝑑))
147		1re 10039	. . . . . . . 8 ⊢ 1 ∈ ℝ
148		elicopnf 12269	. . . . . . . 8 ⊢ (1 ∈ ℝ → (((𝑥↑2) / 𝑑) ∈ (1[,)+∞) ↔ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 1 ≤ ((𝑥↑2) / 𝑑))))
149	147, 148	ax-mp 5	. . . . . . 7 ⊢ (((𝑥↑2) / 𝑑) ∈ (1[,)+∞) ↔ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 1 ≤ ((𝑥↑2) / 𝑑)))
150	144, 146, 149	sylanbrc 698	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ (1[,)+∞))
151		dchrisum0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
152	151	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
153		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (⌊‘𝑦) = (⌊‘((𝑥↑2) / 𝑑)))
154	153	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))))
155	154	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → ((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆))
156	155	fveq2d 6195	. . . . . . . 8 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)))
157		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑑)))
158	157	oveq2d 6666	. . . . . . . 8 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (𝐶 / (√‘𝑦)) = (𝐶 / (√‘((𝑥↑2) / 𝑑))))
159	156, 158	breq12d 4666	. . . . . . 7 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘((𝑥↑2) / 𝑑)))))
160	159	rspcv 3305	. . . . . 6 ⊢ (((𝑥↑2) / 𝑑) ∈ (1[,)+∞) → (∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘((𝑥↑2) / 𝑑)))))
161	150, 152, 160	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘((𝑥↑2) / 𝑑))))
162	132	rpregt0d 11878	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑥) ∈ ℝ ∧ 0 < (√‘𝑥)))
163	140	rpregt0d 11878	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘((𝑥↑2) / 𝑑)) ∈ ℝ ∧ 0 < (√‘((𝑥↑2) / 𝑑))))
164	95	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
165	131	rprege0d 11879	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
166	139	rprege0d 11879	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 0 ≤ ((𝑥↑2) / 𝑑)))
167		sqrtle 14001	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 0 ≤ ((𝑥↑2) / 𝑑))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ↔ (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑))))
168	165, 166, 167	syl2anc 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ↔ (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑))))
169	145, 168	mpbid 222	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑)))
170		lediv2a 10917	. . . . . 6 ⊢ (((((√‘𝑥) ∈ ℝ ∧ 0 < (√‘𝑥)) ∧ ((√‘((𝑥↑2) / 𝑑)) ∈ ℝ ∧ 0 < (√‘((𝑥↑2) / 𝑑))) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑))) → (𝐶 / (√‘((𝑥↑2) / 𝑑))) ≤ (𝐶 / (√‘𝑥)))
171	162, 163, 164, 169, 170	syl31anc 1329	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘((𝑥↑2) / 𝑑))) ≤ (𝐶 / (√‘𝑥)))
172	87, 141, 133, 161, 171	letrd 10194	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘𝑥)))
173	85, 88	abssubd 14192	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)))
174		elicopnf 12269	. . . . . . . 8 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
175	147, 174	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
176	66, 76, 175	sylanbrc 698	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ (1[,)+∞))
177		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (⌊‘𝑦) = (⌊‘𝑥))
178	177	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
179	178	oveq1d 6665	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆))
180	179	fveq2d 6195	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)))
181		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (√‘𝑦) = (√‘𝑥))
182	181	oveq2d 6666	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐶 / (√‘𝑦)) = (𝐶 / (√‘𝑥)))
183	180, 182	breq12d 4666	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)) ≤ (𝐶 / (√‘𝑥))))
184	183	rspcv 3305	. . . . . 6 ⊢ (𝑥 ∈ (1[,)+∞) → (∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)) ≤ (𝐶 / (√‘𝑥))))
185	176, 152, 184	sylc 65	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)) ≤ (𝐶 / (√‘𝑥)))
186	173, 185	eqbrtrd 4675	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) ≤ (𝐶 / (√‘𝑥)))
187	87, 90, 133, 133, 172, 186	le2addd 10646	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ≤ ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
188		2cnd 11093	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
189	96	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
190	189	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
191	190	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
192	100	rpcnne0d 11881	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0))
193	192	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0))
194		divass 10703	. . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0)) → ((2 · 𝐶) / (√‘𝑥)) = (2 · (𝐶 / (√‘𝑥))))
195	188, 191, 193, 194	syl3anc 1326	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) = (2 · (𝐶 / (√‘𝑥))))
196	133	recnd 10068	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘𝑥)) ∈ ℂ)
197	196	2timesd 11275	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (2 · (𝐶 / (√‘𝑥))) = ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
198	195, 197	eqtrd 2656	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) = ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
199	187, 198	breqtrrd 4681	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ≤ ((2 · 𝐶) / (√‘𝑥)))
200	41, 91, 102, 129, 199	letrd 10194	1 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ ((2 · 𝐶) / (√‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ∅c0 3915 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 [,)cico 12177 ...cfz 12326 ⌊cfl 12591 seqcseq 12801 ↑cexp 12860 √csqrt 13973 abscabs 13974 ⇝ cli 14215 Σcsu 14416 Basecbs 15857 0_gc0g 16100 ℤRHomczrh 19848 ℤ/nℤczn 19851 DChrcdchr 24957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-dchr 24958

This theorem is referenced by: dchrisum0lem1 25205

W3C validator