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Theorem dchrisum0lem2a 27657

Description: Lemma for dchrisum0 27660. (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 14008	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝜑)
3		elfznn 13580	. . . . 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 4035	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11	9, 10	sselid 3934	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
12	11	eldifad 3916	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
14		nnz 12611	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
15	14	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
16	4, 5, 6, 7, 13, 15	dchrzrhcl 27385	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
17		nnrp 13027	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
18	17	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
19	18	rpsqrtcld 15462	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
20	19	rpcnd 13061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
21	19	rpne0d 13064	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
22	16, 20, 21	divcld 11990	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
23	2, 3, 22	syl2an 607	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
24	1, 23	fsumcl 15783	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
25		dchrisum0lem2.u	. . . . 5 ⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
26		rlimcl 15553	. . . . 5 ⊢ (𝐻 ⇝_𝑟 𝑈 → 𝑈 ∈ ℂ)
27	25, 26	syl 18	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂ)
28	27	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑈 ∈ ℂ)
29		0xr 11255	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
30		0lt1 11735	. . . . . . . . 9 ⊢ 0 < 1
31		df-ioo 13375	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
32		df-ico 13377	. . . . . . . . . 10 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
33		xrltletr 13181	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
34	31, 32, 33	ixxss1 13389	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
35	29, 30, 34	mp2an 704	. . . . . . . 8 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
36		ioorp 13451	. . . . . . . 8 ⊢ (0(,)+∞) = ℝ⁺
37	35, 36	sseqtri 3984	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
38		resmpt 6039	. . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
41	3	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
42		2fveq3 6886	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
43		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (√‘𝑎) = (√‘𝑚))
44	42, 43	oveq12d 7428	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
45		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
46		ovex 7443	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) ∈ V
47	44, 45, 46	fvmpt3i 6995	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
48	41, 47	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
49	40, 48	sylanl2 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
50		1re 11207	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
51		elicopnf 13471	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
53		flge1nn 13853	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
54	52, 53	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ)
55	54	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ ℕ)
56		nnuz 12900	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
57	55, 56	eleqtrdi 2871	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
58	40, 23	sylanl2 693	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
59	49, 57, 58	fsumser 15780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
60	59	mpteq2dva 5203	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
61	39, 60	eqtrid 2808	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
62		fveq2 6881	. . . . . . 7 ⊢ (𝑚 = (⌊‘𝑥) → (seq1( + , 𝐹)‘𝑚) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
63		rpssre 13023	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
64	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
65	37, 64	sstrid 3947	. . . . . . 7 ⊢ (𝜑 → (1[,)+∞) ⊆ ℝ)
66		1zzd 12624	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
67	44	cbvmptv 5214	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
68	45, 67	eqtri 2784	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
69	22, 68	fmptd 7109	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
70	69	ffvelcdmda 7079	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
71	56, 66, 70	serf 14065	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
72	71	feqmptd 6949	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) = (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)))
73		dchrisum0.s	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
74	72, 73	eqbrtrrd 5134	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)) ⇝ 𝑆)
75	71	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq1( + , 𝐹)‘𝑚) ∈ ℂ)
76	52	simprbi 502	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
77	76	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
78	56, 62, 65, 66, 74, 75, 77	climrlim2 15597	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆)
79		rlimo1 15667	. . . . . 6 ⊢ ((𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ 𝑂(1))
80	78, 79	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ∈ 𝑂(1))
81	61, 80	eqeltrd 2861	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) ∈ 𝑂(1))
82	24	fmpttd 7110	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))):ℝ⁺⟶ℂ)
83		1red 11208	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
84	82, 64, 83	o1resb 15616	. . . 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 260	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ 𝑂(1))
86		o1const 15670	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝑈 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
87	63, 27, 86	sylancr 598	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
88	24, 28, 85, 87	o1mul2 15675	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) ∈ 𝑂(1))
89		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
90		2z 12625	. . . . . . . . 9 ⊢ 2 ∈ ℤ
91		rpexpcl 14115	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
92	89, 90, 91	sylancl 597	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
93	3	nnrpd 13057	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
94		rpdivcl 13042	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
95	92, 93, 94	syl2an 607	. . . . . . 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 27121	. . . . . . . 8 ⊢ 𝐻:ℝ⁺⟶ℝ
98	97	ffvelcdmi 7078	. . . . . . 7 ⊢ (((𝑥↑2) / 𝑚) ∈ ℝ⁺ → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℝ)
99	95, 98	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℝ)
100	99	recnd 11236	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℂ)
101	23, 100	mulcld 11228	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
102	1, 101	fsumcl 15783	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
103	24, 28	mulcld 11228	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
104	25	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝐻 ⇝_𝑟 𝑈)
105	104, 26	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑈 ∈ ℂ)
106	23, 105	mulcld 11228	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
107	1, 101, 106	fsumsub 15838	. . . . . 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 11669	. . . . . . 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 15752	. . . . . 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 15835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈))
111	110	oveq2d 7426	. . . . . 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 2806	. . . . 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 5203	. . . 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 11568	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈) ∈ ℂ)
115	23, 114	mulcld 11228	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
116	1, 115	fsumcl 15783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
117	116	abscld 15489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
118	115	abscld 15489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
119	1, 118	fsumrecl 15784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
120		1red 11208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
121	1, 115	fsumabs 15852	. . . . . . 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 13031	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
123	122	adantl 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
124	123	simpld 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
125		reflcl 13828	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
126	124, 125	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
127	126, 89	rerpdivcld 13090	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
128		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
129	128	rprecred 13070	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
130	23	abscld 15489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
131	93	rpsqrtcld 15462	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → (√‘𝑚) ∈ ℝ⁺)
132	131	adantl 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
133	132	rprecred 13070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℝ)
134	114	abscld 15489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℝ)
135	132, 128	rpdivcld 13076	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ⁺)
136	63, 135	sselid 3934	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ)
137	23	absge0d 15497	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
138	114	absge0d 15497	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)))
139	2, 3, 16	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
140	132	rpcnd 13061	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
141	132	rpne0d 13064	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
142	139, 140, 141	absdivd 15508	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))))
143	132	rprege0d 13066	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)))
144		absid 15346	. . . . . . . . . . . . . . . 16 ⊢ (((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)) → (abs‘(√‘𝑚)) = (√‘𝑚))
145	143, 144	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(√‘𝑚)) = (√‘𝑚))
146	145	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
147	142, 146	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
148	139	abscld 15489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
149		1red 11208	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
150		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑍) = (Base‘𝑍)
151	12	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
152		rpvmasum.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
153	152	nnnn0d 12564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
154	5, 150, 7	znzrhfo 21676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
155		fof 6792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍))
156	153, 154, 155	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍))
157	156	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿:ℤ⟶(Base‘𝑍))
158		elfzelz 13551	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
159		ffvelcdm 7076	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
160	157, 158, 159	syl2an 607	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑚) ∈ (Base‘𝑍))
161	4, 6, 5, 150, 151, 160	dchrabs2 27402	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ≤ 1)
162	148, 149, 132, 161	lediv1dd 13117	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)) ≤ (1 / (√‘𝑚)))
163	147, 162	eqbrtrd 5132	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ (1 / (√‘𝑚)))
164	96, 104	divsqrtsum2 27123	. . . . . . . . . . . . . 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 699	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ (1 / (√‘((𝑥↑2) / 𝑚))))
166	92	rprege0d 13066	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
167		sqrtdiv 15315	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
168	166, 93, 167	syl2an 607	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
169	122	ad2antlr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
170		sqrtsq 15319	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (√‘(𝑥↑2)) = 𝑥)
171	169, 170	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘(𝑥↑2)) = 𝑥)
172	171	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
173	168, 172	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
174	173	oveq2d 7426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = (1 / (𝑥 / (√‘𝑚))))
175		rpcnne0 13034	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
176	175	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	132	rpcnne0d 13068	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
178		recdiv 11920	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
179	176, 177, 178	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
180	174, 179	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = ((√‘𝑚) / 𝑥))
181	165, 180	breqtrd 5136	. . . . . . . . . . . 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 12156	. . . . . . . . . . 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 15507	. . . . . . . . . . 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 11201	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
185		dmdcan 11924	. . . . . . . . . . . . 13 ⊢ ((((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
186	177, 176, 184, 185	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
187	135	rpcnd 13061	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℂ)
188		reccl 11878	. . . . . . . . . . . . . 14 ⊢ (((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) → (1 / (√‘𝑚)) ∈ ℂ)
189	177, 188	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℂ)
190	187, 189	mulcomd 11229	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
191	186, 190	eqtr3d 2798	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
192	182, 183, 191	3brtr4d 5142	. . . . . . . . . 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 15850	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥))
194		flge0nn0 13852	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
195		hashfz1 14381	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
196	123, 194, 195	3syl 19	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
197	196	oveq1d 7425	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
198	89	rpreccld 13069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
199	198	rpcnd 13061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
200		fsumconst 15840	. . . . . . . . . . 11 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
201	1, 199, 200	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
202	126	recnd 11236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℂ)
203	175	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
204	203	simpld 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
205	203	simprd 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
206	202, 204, 205	divrecd 11993	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
207	197, 201, 206	3eqtr4d 2806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
208	193, 207	breqtrd 5136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ ((⌊‘𝑥) / 𝑥))
209		flle 13831	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
210	124, 209	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
211	124	recnd 11236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
212	211	mulridd 11225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 1) = 𝑥)
213	210, 212	breqtrrd 5138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ (𝑥 · 1))
214		rpregt0 13030	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
215	214	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
216		ledivmul 12090	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
217	126, 120, 215, 216	syl3anc 1396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
218	213, 217	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ≤ 1)
219	119, 127, 120, 208, 218	letrd 11366	. . . . . . 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 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
221	220	adantrr 729	. . . . 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 15586	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ 𝑂(1))
223	113, 222	eqeltrrd 2862	. . 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 15680	. 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 260	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 209 ∧ wa 400 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 ∀wral 3077 {crab 3414 ∖ cdif 3901 ⊆ wss 3904 {csn 4588 class class class wbr 5108 ↦ cmpt 5191 ↾ cres 5663 ⟶wf 6532 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7410 Fincfn 8942 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 +∞cpnf 11239 ℝ^*cxr 11241 < clt 11242 ≤ cle 11243 − cmin 11440 / cdiv 11870 ℕcn 12232 2c2 12294 ℕ₀cn0 12503 ℤcz 12590 ℤ_≥cuz 12861 ℝ⁺crp 13015 (,)cioo 13371 [,)cico 13373 ...cfz 13534 ⌊cfl 13822 seqcseq 14036 ↑cexp 14096 ♯chash 14365 √csqrt 15283 abscabs 15284 ⇝ cli 15534 ⇝_𝑟 crli 15535 𝑂(1)co1 15536 Σcsu 15736 Basecbs 17268 0_gc0g 17491 ℤRHomczrh 21628 ℤ/nℤczn 21631 DChrcdchr 27372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-omul 8457 df-er 8693 df-ec 8695 df-qs 8699 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-acn 9927 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13375 df-ioc 13376 df-ico 13377 df-icc 13378 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15103 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-limsup 15521 df-clim 15538 df-rlim 15539 df-o1 15540 df-lo1 15541 df-sum 15737 df-ef 16120 df-sin 16122 df-cos 16123 df-pi 16125 df-dvds 16310 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-qus 17562 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-nsg 19189 df-eqg 19190 df-ghm 19283 df-cntz 19386 df-od 19597 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-rhm 20553 df-subrng 20630 df-subrg 20654 df-drng 20814 df-lmod 20962 df-lss 21032 df-lsp 21072 df-sra 21273 df-rgmod 21274 df-lidl 21311 df-rsp 21312 df-2idl 21368 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-zring 21576 df-zrh 21632 df-zn 21635 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-lp 23272 df-perf 23273 df-cn 23363 df-cnp 23364 df-haus 23451 df-cmp 23523 df-tx 23698 df-hmeo 23891 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-tms 24458 df-cncf 25016 df-limc 26004 df-dv 26005 df-log 26697 df-cxp 26698 df-dchr 27373

This theorem is referenced by: dchrisum0lem2 27658

W3C validator