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Theorem dchrisum0lem1b 27446

Description: Lemma for dchrisum0lem1 27447. (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 13872	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
2		ssun2 4127	. . . . . . 7 ⊢ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
3		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
4	3	rprege0d 12933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
5		flge0nn0 13716	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℕ₀)
7		nn0p1nn 12412	. . . . . . . . . . 11 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
9	8	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ ℕ)
10		nnuz 12767	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
11	9, 10	eleqtrdi 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
12		dchrisum0lem1a 27417	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))))
13	12	simprd 495	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥)))
14		fzsplit2 13441	. . . . . . . 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 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))))
16	2, 15	sseqtrrid 3976	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
17	16	sselda 3932	. . . . 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	22	ssrab3 4030	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
24		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
25	23, 24	sselid 3930	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
26	25	eldifad 3912	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
27	26	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑋 ∈ 𝐷)
28		elfzelz 13416	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℤ)
29	28	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℤ)
30	18, 19, 20, 21, 27, 29	dchrzrhcl 27176	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
31		elfznn 13445	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))) → 𝑚 ∈ ℕ)
32	31	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℕ)
33	32	nnrpd 12924	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → 𝑚 ∈ ℝ⁺)
34	33	rpsqrtcld 15311	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℝ⁺)
35	34	rpcnd 12928	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ∈ ℂ)
36	34	rpne0d 12931	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (√‘𝑚) ≠ 0)
37	30, 35, 36	divcld 11889	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
38	17, 37	syldan 591	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
39	1, 38	fsumcl 15632	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
40	39	abscld 15338	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
41		1zzd 12495	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℤ)
42	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
43		nnz 12481	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
44	43	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
45	18, 19, 20, 21, 42, 44	dchrzrhcl 27176	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
46		nnrp 12894	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
47	46	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
48	47	rpsqrtcld 15311	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
49	48	rpcnd 12928	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
50	48	rpne0d 12931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
51	45, 49, 50	divcld 11889	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
52		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
53		2fveq3 6822	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
54		fveq2 6817	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑚 → (√‘𝑎) = (√‘𝑚))
55	53, 54	oveq12d 7359	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
56	55	cbvmptv 5193	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
57	52, 56	eqtri 2753	. . . . . . . . . 10 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
58	51, 57	fmptd 7042	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
59	58	ffvelcdmda 7012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
60	10, 41, 59	serf 13929	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
61	60	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → seq1( + , 𝐹):ℕ⟶ℂ)
62	3	rpregt0d 12932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
63	62	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
64	63	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
65		1red 11105	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
66		elfznn 13445	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
67	66	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
68	67	nnred 12132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ)
69	67	nnge1d 12165	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑑)
70	3	rpred 12926	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
71		fznnfl 13758	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
72	70, 71	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑑 ∈ (1...(⌊‘𝑥)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝑥)))
73	72	simplbda 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ≤ 𝑥)
74	65, 68, 64, 69, 73	letrd 11262	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ 𝑥)
75		flge1nn 13717	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
76	64, 74, 75	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ ℕ)
77		eluznn 12808	. . . . . . 7 ⊢ (((⌊‘𝑥) ∈ ℕ ∧ (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℕ)
78	76, 13, 77	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ ℕ)
79	61, 78	ffvelcdmd 7013	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) ∈ ℂ)
80		dchrisum0.s	. . . . . . 7 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
81		climcl 15398	. . . . . . 7 ⊢ (seq1( + , 𝐹) ⇝ 𝑆 → 𝑆 ∈ ℂ)
82	80, 81	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ℂ)
83	82	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑆 ∈ ℂ)
84	79, 83	subcld 11464	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆) ∈ ℂ)
85	84	abscld 15338	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ∈ ℝ)
86	61, 76	ffvelcdmd 7013	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (seq1( + , 𝐹)‘(⌊‘𝑥)) ∈ ℂ)
87	83, 86	subcld 11464	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ ℂ)
88	87	abscld 15338	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) ∈ ℝ)
89	85, 88	readdcld 11133	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ∈ ℝ)
90		2re 12191	. . . . . 6 ⊢ 2 ∈ ℝ
91		dchrisum0.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
92		elrege0 13346	. . . . . . . 8 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
93	91, 92	sylib 218	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
94	93	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℝ)
95		remulcl 11083	. . . . . 6 ⊢ ((2 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (2 · 𝐶) ∈ ℝ)
96	90, 94, 95	sylancr 587	. . . . 5 ⊢ (𝜑 → (2 · 𝐶) ∈ ℝ)
97	96	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (2 · 𝐶) ∈ ℝ)
98	3	rpsqrtcld 15311	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (√‘𝑥) ∈ ℝ⁺)
99	97, 98	rerpdivcld 12957	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
100	99	adantr 480	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) ∈ ℝ)
101		ssun1 4126	. . . . . . . . . . 11 ⊢ (1...(⌊‘𝑥)) ⊆ ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑))))
102	101, 15	sseqtrrid 3976	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘((𝑥↑2) / 𝑑))))
103	102	sselda 3932	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑))))
104		ovex 7374	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) ∈ V
105	55, 52, 104	fvmpt3i 6929	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
106	32, 105	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
107	103, 106	syldan 591	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
108	76, 10	eleqtrdi 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
109	103, 37	syldan 591	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
110	107, 108, 109	fsumser 15629	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
111	110, 86	eqeltrd 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
112	111, 39	pncan2d 11466	. . . . 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)))
113		reflcl 13692	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
114	64, 113	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) ∈ ℝ)
115	114	ltp1d 12044	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘𝑥) < ((⌊‘𝑥) + 1))
116		fzdisj 13443	. . . . . . . . 9 ⊢ ((⌊‘𝑥) < ((⌊‘𝑥) + 1) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) = ∅)
117	115, 116	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘((𝑥↑2) / 𝑑)))) = ∅)
118		fzfid 13872	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘((𝑥↑2) / 𝑑))) ∈ Fin)
119	117, 15, 118, 37	fsumsplit 15640	. . . . . . 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))))
120	78, 10	eleqtrdi 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (⌊‘((𝑥↑2) / 𝑑)) ∈ (ℤ_≥‘1))
121	106, 120, 37	fsumser 15629	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘((𝑥↑2) / 𝑑)))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))))
122	119, 121	eqtr3d 2767	. . . . . 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))))
123	122, 110	oveq12d 7359	. . . . 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))))
124	112, 123	eqtr3d 2767	. . . 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))))
125	124	fveq2d 6821	. . 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)))))
126	79, 86, 83	abs3difd 15362	. . 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))))))
127	125, 126	eqbrtrd 5111	. 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))))))
128	94	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℝ)
129		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
130	129	rpsqrtcld 15311	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑥) ∈ ℝ⁺)
131	128, 130	rerpdivcld 12957	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘𝑥)) ∈ ℝ)
132		2z 12496	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
133		rpexpcl 13979	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
134	3, 132, 133	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
135	134	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥↑2) ∈ ℝ⁺)
136	67	nnrpd 12924	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
137	135, 136	rpdivcld 12943	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ ℝ⁺)
138	137	rpsqrtcld 15311	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑑)) ∈ ℝ⁺)
139	128, 138	rerpdivcld 12957	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘((𝑥↑2) / 𝑑))) ∈ ℝ)
140		2fveq3 6822	. . . . . . . 8 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))))
141	140	fvoveq1d 7363	. . . . . . 7 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)))
142		fveq2 6817	. . . . . . . 8 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑑)))
143	142	oveq2d 7357	. . . . . . 7 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → (𝐶 / (√‘𝑦)) = (𝐶 / (√‘((𝑥↑2) / 𝑑))))
144	141, 143	breq12d 5102	. . . . . 6 ⊢ (𝑦 = ((𝑥↑2) / 𝑑) → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘((𝑥↑2) / 𝑑)))))
145		dchrisum0.1	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
146	145	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
147	134	rpred 12926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ)
148		nndivre 12158	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ ∧ 𝑑 ∈ ℕ) → ((𝑥↑2) / 𝑑) ∈ ℝ)
149	147, 66, 148	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ ℝ)
150	12	simpld 494	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ≤ ((𝑥↑2) / 𝑑))
151	65, 64, 149, 74, 150	letrd 11262	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 1 ≤ ((𝑥↑2) / 𝑑))
152		1re 11104	. . . . . . . 8 ⊢ 1 ∈ ℝ
153		elicopnf 13337	. . . . . . . 8 ⊢ (1 ∈ ℝ → (((𝑥↑2) / 𝑑) ∈ (1[,)+∞) ↔ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 1 ≤ ((𝑥↑2) / 𝑑))))
154	152, 153	ax-mp 5	. . . . . . 7 ⊢ (((𝑥↑2) / 𝑑) ∈ (1[,)+∞) ↔ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 1 ≤ ((𝑥↑2) / 𝑑)))
155	149, 151, 154	sylanbrc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑑) ∈ (1[,)+∞))
156	144, 146, 155	rspcdva 3576	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘((𝑥↑2) / 𝑑))))
157	130	rpregt0d 12932	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑥) ∈ ℝ ∧ 0 < (√‘𝑥)))
158	138	rpregt0d 12932	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘((𝑥↑2) / 𝑑)) ∈ ℝ ∧ 0 < (√‘((𝑥↑2) / 𝑑))))
159	93	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
160	129	rprege0d 12933	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
161	137	rprege0d 12933	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 0 ≤ ((𝑥↑2) / 𝑑)))
162		sqrtle 15159	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) ∧ (((𝑥↑2) / 𝑑) ∈ ℝ ∧ 0 ≤ ((𝑥↑2) / 𝑑))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ↔ (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑))))
163	160, 161, 162	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 ≤ ((𝑥↑2) / 𝑑) ↔ (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑))))
164	150, 163	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (√‘𝑥) ≤ (√‘((𝑥↑2) / 𝑑)))
165		lediv2a 12008	. . . . . 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)))
166	157, 158, 159, 164, 165	syl31anc 1375	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘((𝑥↑2) / 𝑑))) ≤ (𝐶 / (√‘𝑥)))
167	85, 139, 131, 156, 166	letrd 11262	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) ≤ (𝐶 / (√‘𝑥)))
168	83, 86	abssubd 15355	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)))
169		2fveq3 6822	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
170	169	fvoveq1d 7363	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)))
171		fveq2 6817	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (√‘𝑦) = (√‘𝑥))
172	171	oveq2d 7357	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐶 / (√‘𝑦)) = (𝐶 / (√‘𝑥)))
173	170, 172	breq12d 5102	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)) ≤ (𝐶 / (√‘𝑥))))
174		elicopnf 13337	. . . . . . . 8 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
175	152, 174	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
176	64, 74, 175	sylanbrc 583	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ (1[,)+∞))
177	173, 146, 176	rspcdva 3576	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑆)) ≤ (𝐶 / (√‘𝑥)))
178	168, 177	eqbrtrd 5111	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥)))) ≤ (𝐶 / (√‘𝑥)))
179	85, 88, 131, 131, 167, 178	le2addd 11728	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ≤ ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
180		2cnd 12195	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
181	94	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
182	181	recnd 11132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐶 ∈ ℂ)
183	182	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝐶 ∈ ℂ)
184	98	rpcnne0d 12935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0))
185	184	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0))
186		divass 11786	. . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ((√‘𝑥) ∈ ℂ ∧ (√‘𝑥) ≠ 0)) → ((2 · 𝐶) / (√‘𝑥)) = (2 · (𝐶 / (√‘𝑥))))
187	180, 183, 185, 186	syl3anc 1373	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) = (2 · (𝐶 / (√‘𝑥))))
188	131	recnd 11132	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝐶 / (√‘𝑥)) ∈ ℂ)
189	188	2timesd 12356	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (2 · (𝐶 / (√‘𝑥))) = ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
190	187, 189	eqtrd 2765	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((2 · 𝐶) / (√‘𝑥)) = ((𝐶 / (√‘𝑥)) + (𝐶 / (√‘𝑥))))
191	179, 190	breqtrrd 5117	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((abs‘((seq1( + , 𝐹)‘(⌊‘((𝑥↑2) / 𝑑))) − 𝑆)) + (abs‘(𝑆 − (seq1( + , 𝐹)‘(⌊‘𝑥))))) ≤ ((2 · 𝐶) / (√‘𝑥)))
192	40, 89, 100, 127, 191	letrd 11262	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 ∀wral 3045 {crab 3393 ∖ cdif 3897 ∪ cun 3898 ∩ cin 3899 ∅c0 4281 {csn 4574 class class class wbr 5089 ↦ cmpt 5170 ⟶wf 6473 ‘cfv 6477 (class class class)co 7341 ℂcc 10996 ℝcr 10997 0cc0 10998 1c1 10999 + caddc 11001 · cmul 11003 +∞cpnf 11135 < clt 11138 ≤ cle 11139 − cmin 11336 / cdiv 11766 ℕcn 12117 2c2 12172 ℕ₀cn0 12373 ℤcz 12460 ℤ_≥cuz 12724 ℝ⁺crp 12882 [,)cico 13239 ...cfz 13399 ⌊cfl 13686 seqcseq 13900 ↑cexp 13960 √csqrt 15132 abscabs 15133 ⇝ cli 15383 Σcsu 15585 Basecbs 17112 0_gc0g 17335 ℤRHomczrh 21429 ℤ/nℤczn 21432 DChrcdchr 27163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-inf2 9526 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076 ax-addf 11077 ax-mulf 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-isom 6486 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-ec 8619 df-qs 8623 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9824 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-rp 12883 df-ico 13243 df-fz 13400 df-fzo 13547 df-fl 13688 df-seq 13901 df-exp 13961 df-hash 14230 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-clim 15387 df-sum 15586 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-starv 17168 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-unif 17176 df-0g 17337 df-imas 17404 df-qus 17405 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-grp 18841 df-minusg 18842 df-sbg 18843 df-mulg 18973 df-subg 19028 df-nsg 19029 df-eqg 19030 df-ghm 19118 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-cring 20147 df-oppr 20248 df-dvdsr 20268 df-unit 20269 df-rhm 20383 df-subrng 20454 df-subrg 20478 df-lmod 20788 df-lss 20858 df-lsp 20898 df-sra 21100 df-rgmod 21101 df-lidl 21138 df-rsp 21139 df-2idl 21180 df-cnfld 21285 df-zring 21377 df-zrh 21433 df-zn 21436 df-dchr 27164

This theorem is referenced by: dchrisum0lem1 27447

W3C validator