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Theorem dchrisum0lem2a 27485

Description: Lemma for dchrisum0 27488. (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 13996	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝜑)
3		elfznn 13575	. . . . 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 4062	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11	9, 10	sselid 3961	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
12	11	eldifad 3943	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
14		nnz 12614	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
16	4, 5, 6, 7, 13, 15	dchrzrhcl 27213	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
17		nnrp 13025	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
19	18	rpsqrtcld 15435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
20	19	rpcnd 13058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
21	19	rpne0d 13061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
22	16, 20, 21	divcld 12022	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
23	2, 3, 22	syl2an 596	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
24	1, 23	fsumcl 15754	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
25		dchrisum0lem2.u	. . . . 5 ⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
26		rlimcl 15524	. . . . 5 ⊢ (𝐻 ⇝_𝑟 𝑈 → 𝑈 ∈ ℂ)
27	25, 26	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂ)
28	27	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑈 ∈ ℂ)
29		0xr 11287	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
30		0lt1 11764	. . . . . . . . 9 ⊢ 0 < 1
31		df-ioo 13371	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
32		df-ico 13373	. . . . . . . . . 10 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
33		xrltletr 13178	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
34	31, 32, 33	ixxss1 13385	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
35	29, 30, 34	mp2an 692	. . . . . . . 8 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
36		ioorp 13447	. . . . . . . 8 ⊢ (0(,)+∞) = ℝ⁺
37	35, 36	sseqtri 4012	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
38		resmpt 6029	. . . . . . 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 3959	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
41	3	adantl 481	. . . . . . . . . 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 6996	. . . . . . . . . 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 681	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
50		1re 11240	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
51		elicopnf 13467	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
53		flge1nn 13843	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
54	52, 53	sylbi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ)
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ ℕ)
56		nnuz 12900	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
57	55, 56	eleqtrdi 2845	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
58	40, 23	sylanl2 681	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
59	49, 57, 58	fsumser 15751	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
60	59	mpteq2dva 5219	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
61	39, 60	eqtrid 2783	. . . . 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 13021	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
64	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
65	37, 64	sstrid 3975	. . . . . . 7 ⊢ (𝜑 → (1[,)+∞) ⊆ ℝ)
66		1zzd 12628	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
67	44	cbvmptv 5230	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
68	45, 67	eqtri 2759	. . . . . . . . . . . 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 14053	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
72	71	feqmptd 6952	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) = (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)))
73		dchrisum0.s	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
74	72, 73	eqbrtrrd 5148	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)) ⇝ 𝑆)
75	71	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (seq1( + , 𝐹)‘𝑚) ∈ ℂ)
76	52	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (1[,)+∞) → 1 ≤ 𝑥)
77	76	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
78	56, 62, 65, 66, 74, 75, 77	climrlim2 15568	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆)
79		rlimo1 15638	. . . . . 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 2835	. . . 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 11241	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
84	82, 64, 83	o1resb 15587	. . . 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 257	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ 𝑂(1))
86		o1const 15641	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝑈 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
87	63, 27, 86	sylancr 587	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
88	24, 28, 85, 87	o1mul2 15646	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈)) ∈ 𝑂(1))
89		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
90		2z 12629	. . . . . . . . 9 ⊢ 2 ∈ ℤ
91		rpexpcl 14103	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
92	89, 90, 91	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
93	3	nnrpd 13054	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
94		rpdivcl 13039	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
95	92, 93, 94	syl2an 596	. . . . . . 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 26948	. . . . . . . 8 ⊢ 𝐻:ℝ⁺⟶ℝ
98	97	ffvelcdmi 7078	. . . . . . 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 11268	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℂ)
101	23, 100	mulcld 11260	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
102	1, 101	fsumcl 15754	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
103	24, 28	mulcld 11260	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
104	25	ad2antrr 726	. . . . . . . . 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 11260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
107	1, 101, 106	fsumsub 15809	. . . . . 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 11698	. . . . . . 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 15723	. . . . . 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 15806	. . . . . . 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 2781	. . . . 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 5219	. . . 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 11599	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈) ∈ ℂ)
115	23, 114	mulcld 11260	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
116	1, 115	fsumcl 15754	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
117	116	abscld 15460	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
118	115	abscld 15460	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
119	1, 118	fsumrecl 15755	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
120		1red 11241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
121	1, 115	fsumabs 15822	. . . . . . 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 13029	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
123	122	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
124	123	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
125		reflcl 13818	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
126	124, 125	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
127	126, 89	rerpdivcld 13087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
128		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
129	128	rprecred 13067	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
130	23	abscld 15460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
131	93	rpsqrtcld 15435	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → (√‘𝑚) ∈ ℝ⁺)
132	131	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
133	132	rprecred 13067	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℝ)
134	114	abscld 15460	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℝ)
135	132, 128	rpdivcld 13073	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ⁺)
136	63, 135	sselid 3961	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ)
137	23	absge0d 15468	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
138	114	absge0d 15468	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)))
139	2, 3, 16	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
140	132	rpcnd 13058	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
141	132	rpne0d 13061	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
142	139, 140, 141	absdivd 15479	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))))
143	132	rprege0d 13063	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)))
144		absid 15320	. . . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
147	142, 146	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
148	139	abscld 15460	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
149		1red 11241	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
150		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑍) = (Base‘𝑍)
151	12	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
152		rpvmasum.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
153	152	nnnn0d 12567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
154	5, 150, 7	znzrhfo 21513	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
155		fof 6795	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐿:ℤ⟶(Base‘𝑍))
158		elfzelz 13546	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑚) ∈ (Base‘𝑍))
161	4, 6, 5, 150, 151, 160	dchrabs2 27230	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ≤ 1)
162	148, 149, 132, 161	lediv1dd 13114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)) ≤ (1 / (√‘𝑚)))
163	147, 162	eqbrtrd 5146	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ (1 / (√‘𝑚)))
164	96, 104	divsqrtsum2 26950	. . . . . . . . . . . . . 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 687	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ (1 / (√‘((𝑥↑2) / 𝑚))))
166	92	rprege0d 13063	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
167		sqrtdiv 15289	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
168	166, 93, 167	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
169	122	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
170		sqrtsq 15293	. . . . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
173	168, 172	eqtrd 2771	. . . . . . . . . . . . . . 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 13032	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
176	175	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	132	rpcnne0d 13065	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
178		recdiv 11952	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
179	176, 177, 178	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
180	174, 179	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = ((√‘𝑚) / 𝑥))
181	165, 180	breqtrd 5150	. . . . . . . . . . . 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 12189	. . . . . . . . . . 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 15478	. . . . . . . . . . 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 11235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
185		dmdcan 11956	. . . . . . . . . . . . 13 ⊢ ((((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
186	177, 176, 184, 185	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
187	135	rpcnd 13058	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℂ)
188		reccl 11908	. . . . . . . . . . . . . 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 11261	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
191	186, 190	eqtr3d 2773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
192	182, 183, 191	3brtr4d 5156	. . . . . . . . . 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 15820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥))
194		flge0nn0 13842	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
195		hashfz1 14369	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
196	123, 194, 195	3syl 18	. . . . . . . . . . 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 13066	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
199	198	rpcnd 13058	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
200		fsumconst 15811	. . . . . . . . . . 11 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
201	1, 199, 200	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
202	126	recnd 11268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℂ)
203	175	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
204	203	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
205	203	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
206	202, 204, 205	divrecd 12025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
207	197, 201, 206	3eqtr4d 2781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
208	193, 207	breqtrd 5150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ ((⌊‘𝑥) / 𝑥))
209		flle 13821	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
210	124, 209	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
211	124	recnd 11268	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
212	211	mulridd 11257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 1) = 𝑥)
213	210, 212	breqtrrd 5152	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ (𝑥 · 1))
214		rpregt0 13028	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
215	214	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
216		ledivmul 12123	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
217	126, 120, 215, 216	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
218	213, 217	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ≤ 1)
219	119, 127, 120, 208, 218	letrd 11397	. . . . . . 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 11397	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
221	220	adantrr 717	. . . . 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 15557	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ 𝑂(1))
223	113, 222	eqeltrrd 2836	. . 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 15651	. 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 257	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 {crab 3420 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 class class class wbr 5124 ↦ cmpt 5206 ↾ cres 5661 ⟶wf 6532 –onto→wfo 6534 ‘cfv 6536 (class class class)co 7410 Fincfn 8964 ℂcc 11132 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 · cmul 11139 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 / cdiv 11899 ℕcn 12245 2c2 12300 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ℝ⁺crp 13013 (,)cioo 13367 [,)cico 13369 ...cfz 13529 ⌊cfl 13812 seqcseq 14024 ↑cexp 14084 ♯chash 14353 √csqrt 15257 abscabs 15258 ⇝ cli 15505 ⇝_𝑟 crli 15506 𝑂(1)co1 15507 Σcsu 15707 Basecbs 17233 0_gc0g 17458 ℤRHomczrh 21465 ℤ/nℤczn 21468 DChrcdchr 27200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 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 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-o1 15511 df-lo1 15512 df-sum 15708 df-ef 16088 df-sin 16090 df-cos 16091 df-pi 16093 df-dvds 16278 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-qus 17528 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-cntz 19305 df-od 19514 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-sra 21136 df-rgmod 21137 df-lidl 21174 df-rsp 21175 df-2idl 21216 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-zn 21472 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25824 df-dv 25825 df-log 26522 df-cxp 26523 df-dchr 27201

This theorem is referenced by: dchrisum0lem2 27486

W3C validator