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Theorem dchrisum0lem2a 27547

Description: Lemma for dchrisum0 27550. (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)

**Assertion**
Ref	Expression
dchrisum0lem2a	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))

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 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) 𝑈(𝑦,𝑎,𝑑) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑚,𝑎,𝑑) 𝐻(𝑥,𝑦,𝑚,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑊(𝑦,𝑚,𝑎,𝑑) 𝑍(𝑎,𝑑)

Proof of Theorem dchrisum0lem2a Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 13972	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝜑)
3		elfznn 13544	. . . . 5 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℕ)
4		rpvmasum2.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
5		rpvmasum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
6		rpvmasum2.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
7		rpvmasum.l	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑍)
8		rpvmasum2.w	. . . . . . . . . . 11 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
9	8	ssrab3 4026	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11	9, 10	sselid 3925	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
12	11	eldifad 3907	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
14		nnz 12575	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
15	14	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
16	4, 5, 6, 7, 13, 15	dchrzrhcl 27275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
17		nnrp 12991	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
18	17	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
19	18	rpsqrtcld 15411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
20	19	rpcnd 13025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
21	19	rpne0d 13028	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
22	16, 20, 21	divcld 11953	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
23	2, 3, 22	syl2an 604	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
24	1, 23	fsumcl 15732	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
25		dchrisum0lem2.u	. . . . 5 ⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
26		rlimcl 15502	. . . . 5 ⊢ (𝐻 ⇝_𝑟 𝑈 → 𝑈 ∈ ℂ)
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂ)
28	27	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑈 ∈ ℂ)
29		0xr 11215	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
30		0lt1 11695	. . . . . . . . 9 ⊢ 0 < 1
31		df-ioo 13339	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
32		df-ico 13341	. . . . . . . . . 10 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
33		xrltletr 13145	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
34	31, 32, 33	ixxss1 13353	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
35	29, 30, 34	mp2an 700	. . . . . . . 8 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
36		ioorp 13415	. . . . . . . 8 ⊢ (0(,)+∞) = ℝ⁺
37	35, 36	sseqtri 3975	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
38		resmpt 6012	. . . . . . 7 ⊢ ((1[,)+∞) ⊆ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
39	37, 38	ax-mp 5	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
40	37	sseli 3923	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
41	3	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
42		2fveq3 6857	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
43		fveq2 6852	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (√‘𝑎) = (√‘𝑚))
44	42, 43	oveq12d 7399	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
45		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
46		ovex 7414	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) ∈ V
47	44, 45, 46	fvmpt3i 6966	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
48	41, 47	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
49	40, 48	sylanl2 689	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
50		1re 11167	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
51		elicopnf 13435	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
53		flge1nn 13817	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
54	52, 53	sylbi 219	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ)
55	54	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ ℕ)
56		nnuz 12864	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
57	55, 56	eleqtrdi 2862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
58	40, 23	sylanl2 689	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
59	49, 57, 58	fsumser 15729	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
60	59	mpteq2dva 5183	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
61	39, 60	eqtrid 2799	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
62		fveq2 6852	. . . . . . 7 ⊢ (𝑚 = (⌊‘𝑥) → (seq1( + , 𝐹)‘𝑚) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
63		rpssre 12987	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
64	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
65	37, 64	sstrid 3938	. . . . . . 7 ⊢ (𝜑 → (1[,)+∞) ⊆ ℝ)
66		1zzd 12588	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
67	44	cbvmptv 5194	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
68	45, 67	eqtri 2775	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
69	22, 68	fmptd 7080	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
70	69	ffvelcdmda 7050	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
71	56, 66, 70	serf 14029	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
72	71	feqmptd 6920	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) = (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)))
73		dchrisum0.s	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
74	72, 73	eqbrtrrd 5114	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)) ⇝ 𝑆)
75	71	ffvelcdmda 7050	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq1( + , 𝐹)‘𝑚) ∈ ℂ)
76	52	simprbi 500	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
77	76	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
78	56, 62, 65, 66, 74, 75, 77	climrlim2 15546	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆)
79		rlimo1 15616	. . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ 𝑂(1))
80	78, 79	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ 𝑂(1))
81	61, 80	eqeltrd 2852	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) ∈ 𝑂(1))
82	24	fmpttd 7081	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))):ℝ⁺⟶ℂ)
83		1red 11168	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
84	82, 64, 83	o1resb 15565	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) ∈ 𝑂(1)))
85	81, 84	mpbird 259	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ 𝑂(1))
86		o1const 15619	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝑈 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
87	63, 27, 86	sylancr 595	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
88	24, 28, 85, 87	o1mul2 15624	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) ∈ 𝑂(1))
89		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
90		2z 12589	. . . . . . . . 9 ⊢ 2 ∈ ℤ
91		rpexpcl 14079	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
92	89, 90, 91	sylancl 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
93	3	nnrpd 13021	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
94		rpdivcl 13006	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
95	92, 93, 94	syl2an 604	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
96		dchrisum0lem2.h	. . . . . . . . 9 ⊢ 𝐻 = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
97	96	divsqrsumf 27011	. . . . . . . 8 ⊢ 𝐻:ℝ⁺⟶ℝ
98	97	ffvelcdmi 7049	. . . . . . 7 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ⁺ → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℝ)
99	95, 98	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℝ)
100	99	recnd 11196	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℂ)
101	23, 100	mulcld 11188	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
102	1, 101	fsumcl 15732	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
103	24, 28	mulcld 11188	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
104	25	ad2antrr 734	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝐻 ⇝_𝑟 𝑈)
105	104, 26	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑈 ∈ ℂ)
106	23, 105	mulcld 11188	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
107	1, 101, 106	fsumsub 15787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) = (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)))
108	23, 100, 105	subdid 11629	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) = ((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)))
109	108	sumeq2dv 15701	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) = Σ𝑚 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)))
110	1, 28, 23	fsummulc1 15784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈))
111	110	oveq2d 7397	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) = (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)))
112	107, 109, 111	3eqtr4d 2797	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) = (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)))
113	112	mpteq2dva 5183	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈))))
114	100, 105	subcld 11528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈) ∈ ℂ)
115	23, 114	mulcld 11188	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
116	1, 115	fsumcl 15732	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
117	116	abscld 15438	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
118	115	abscld 15438	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
119	1, 118	fsumrecl 15733	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
120		1red 11168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
121	1, 115	fsumabs 15801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))))
122		rprege0 12995	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
123	122	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
124	123	simpld 497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
125		reflcl 13792	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
126	124, 125	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
127	126, 89	rerpdivcld 13054	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
128		simplr 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
129	128	rprecred 13034	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
130	23	abscld 15438	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
131	93	rpsqrtcld 15411	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → (√‘𝑚) ∈ ℝ⁺)
132	131	adantl 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
133	132	rprecred 13034	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℝ)
134	114	abscld 15438	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℝ)
135	132, 128	rpdivcld 13040	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ⁺)
136	63, 135	sselid 3925	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ)
137	23	absge0d 15446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
138	114	absge0d 15446	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)))
139	2, 3, 16	syl2an 604	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
140	132	rpcnd 13025	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
141	132	rpne0d 13028	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
142	139, 140, 141	absdivd 15457	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))))
143	132	rprege0d 13030	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)))
144		absid 15295	. . . . . . . . . . . . . . . 16 ⊢ (((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)) → (abs‘(√‘𝑚)) = (√‘𝑚))
145	143, 144	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(√‘𝑚)) = (√‘𝑚))
146	145	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
147	142, 146	eqtrd 2787	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
148	139	abscld 15438	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
149		1red 11168	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
150		eqid 2752	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑍) = (Base‘𝑍)
151	12	ad2antrr 734	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
152		rpvmasum.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
153	152	nnnn0d 12528	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
154	5, 150, 7	znzrhfo 21568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
155		fof 6763	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
156	153, 154, 155	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
157	156	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿:ℤ⟶(Base‘𝑍))
158		elfzelz 13515	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
159		ffvelcdm 7047	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
160	157, 158, 159	syl2an 604	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑚) ∈ (Base‘𝑍))
161	4, 6, 5, 150, 151, 160	dchrabs2 27292	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ≤ 1)
162	148, 149, 132, 161	lediv1dd 13081	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)) ≤ (1 / (√‘𝑚)))
163	147, 162	eqbrtrd 5112	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ (1 / (√‘𝑚)))
164	96, 104	divsqrtsum2 27013	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) ∧ ((𝑥↑2) / 𝑚) ∈ ℝ⁺) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ (1 / (√‘((𝑥↑2) / 𝑚))))
165	95, 164	mpdan 695	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ (1 / (√‘((𝑥↑2) / 𝑚))))
166	92	rprege0d 13030	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
167		sqrtdiv 15264	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
168	166, 93, 167	syl2an 604	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
169	122	ad2antlr 735	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
170		sqrtsq 15268	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘(𝑥↑2)) = 𝑥)
171	169, 170	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘(𝑥↑2)) = 𝑥)
172	171	oveq1d 7396	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
173	168, 172	eqtrd 2787	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
174	173	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = (1 / (𝑥 / (√‘𝑚))))
175		rpcnne0 12998	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
176	175	ad2antlr 735	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	132	rpcnne0d 13032	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
178		recdiv 11883	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
179	176, 177, 178	syl2anc 592	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
180	174, 179	eqtrd 2787	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = ((√‘𝑚) / 𝑥))
181	165, 180	breqtrd 5116	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ ((√‘𝑚) / 𝑥))
182	130, 133, 134, 136, 137, 138, 163, 181	lemul12ad 12120	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) · (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
183	23, 114	absmuld 15456	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) = ((abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) · (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))))
184		1cnd 11161	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
185		dmdcan 11887	. . . . . . . . . . . . 13 ⊢ ((((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
186	177, 176, 184, 185	syl3anc 1382	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
187	135	rpcnd 13025	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℂ)
188		reccl 11838	. . . . . . . . . . . . . 14 ⊢ (((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) → (1 / (√‘𝑚)) ∈ ℂ)
189	177, 188	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℂ)
190	187, 189	mulcomd 11189	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
191	186, 190	eqtr3d 2789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
192	182, 183, 191	3brtr4d 5122	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ (1 / 𝑥))
193	1, 118, 129, 192	fsumle 15799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥))
194		flge0nn0 13816	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
195		hashfz1 14345	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
196	123, 194, 195	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
197	196	oveq1d 7396	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
198	89	rpreccld 13033	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
199	198	rpcnd 13025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
200		fsumconst 15789	. . . . . . . . . . 11 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
201	1, 199, 200	syl2anc 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
202	126	recnd 11196	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℂ)
203	175	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
204	203	simpld 497	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
205	203	simprd 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
206	202, 204, 205	divrecd 11956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
207	197, 201, 206	3eqtr4d 2797	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
208	193, 207	breqtrd 5116	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ ((⌊‘𝑥) / 𝑥))
209		flle 13795	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
210	124, 209	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
211	124	recnd 11196	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
212	211	mulridd 11185	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 1) = 𝑥)
213	210, 212	breqtrrd 5118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ (𝑥 · 1))
214		rpregt0 12994	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
215	214	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
216		ledivmul 12054	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
217	126, 120, 215, 216	syl3anc 1382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
218	213, 217	mpbird 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ≤ 1)
219	119, 127, 120, 208, 218	letrd 11326	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
220	117, 119, 120, 121, 219	letrd 11326	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
221	220	adantrr 725	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
222	64, 116, 83, 83, 221	elo1d 15535	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ 𝑂(1))
223	113, 222	eqeltrrd 2853	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) − (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈))) ∈ 𝑂(1))
224	102, 103, 223	o1dif 15629	. 2 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) ∈ 𝑂(1)))
225	88, 224	mpbird 259	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 ∀wral 3066 {crab 3404 ∖ cdif 3892 ⊆ wss 3895 {csn 4572 class class class wbr 5090 ↦ cmpt 5171 ↾ cres 5638 ⟶wf 6502 –onto→wfo 6504 ‘cfv 6506 (class class class)co 7381 Fincfn 8912 ℂcc 11057 ℝcr 11058 0cc0 11059 1c1 11060 + caddc 11062 · cmul 11064 +∞cpnf 11199 ℝ^*cxr 11201 < clt 11202 ≤ cle 11203 − cmin 11400 / cdiv 11830 ℕcn 12196 2c2 12258 ℕ₀cn0 12467 ℤcz 12554 ℤ_≥cuz 12825 ℝ⁺crp 12979 (,)cioo 13335 [,)cico 13337 ...cfz 13498 ⌊cfl 13786 seqcseq 14000 ↑cexp 14060 ♯chash 14329 √csqrt 15232 abscabs 15233 ⇝ cli 15483 ⇝_𝑟 crli 15484 𝑂(1)co1 15485 Σcsu 15685 Basecbs 17217 0_gc0g 17440 ℤRHomczrh 21520 ℤ/nℤczn 21523 DChrcdchr 27262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5058 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-tpos 8190 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-oadd 8425 df-omul 8426 df-er 8662 df-ec 8664 df-qs 8668 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-acn 9886 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ioc 13340 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-mod 13866 df-seq 14001 df-exp 14061 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15066 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-limsup 15470 df-clim 15487 df-rlim 15488 df-o1 15489 df-lo1 15490 df-sum 15686 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-dvds 16259 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-qus 17511 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-mhm 18789 df-submnd 18790 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-nsg 19138 df-eqg 19139 df-ghm 19226 df-cntz 19329 df-od 19540 df-cmn 19794 df-abl 19795 df-mgp 20159 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20354 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20564 df-subrg 20588 df-drng 20749 df-lmod 20898 df-lss 20968 df-lsp 21008 df-sra 21209 df-rgmod 21210 df-lidl 21247 df-rsp 21248 df-2idl 21289 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-zring 21468 df-zrh 21524 df-zn 21527 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cncf 24909 df-limc 25897 df-dv 25898 df-log 26587 df-cxp 26588 df-dchr 27263

This theorem is referenced by: dchrisum0lem2 27548

W3C validator