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Theorem dchrisum0lem2a 26570

Description: Lemma for dchrisum0 26573. (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 13621	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝜑)
3		elfznn 13214	. . . . 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 4011	. . . . . . . . . 10 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
10		dchrisum0.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
11	9, 10	sselid 3915	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
12	11	eldifad 3895	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
14		nnz 12272	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
15	14	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
16	4, 5, 6, 7, 13, 15	dchrzrhcl 26298	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
17		nnrp 12670	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ⁺)
18	17	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
19	18	rpsqrtcld 15051	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℝ⁺)
20	19	rpcnd 12703	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ∈ ℂ)
21	19	rpne0d 12706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (√‘𝑚) ≠ 0)
22	16, 20, 21	divcld 11681	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
23	2, 3, 22	syl2an 595	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
24	1, 23	fsumcl 15373	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
25		dchrisum0lem2.u	. . . . 5 ⊢ (𝜑 → 𝐻 ⇝_𝑟 𝑈)
26		rlimcl 15140	. . . . 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 10953	. . . . . . . . 9 ⊢ 0 ∈ ℝ^*
30		0lt1 11427	. . . . . . . . 9 ⊢ 0 < 1
31		df-ioo 13012	. . . . . . . . . 10 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
32		df-ico 13014	. . . . . . . . . 10 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
33		xrltletr 12820	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
34	31, 32, 33	ixxss1 13026	. . . . . . . . 9 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
35	29, 30, 34	mp2an 688	. . . . . . . 8 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
36		ioorp 13086	. . . . . . . 8 ⊢ (0(,)+∞) = ℝ⁺
37	35, 36	sseqtri 3953	. . . . . . 7 ⊢ (1[,)+∞) ⊆ ℝ⁺
38		resmpt 5934	. . . . . . 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 3913	. . . . . . . . 9 ⊢ (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ⁺)
41	3	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑚 ∈ ℕ)
42		2fveq3 6761	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
43		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑚 → (√‘𝑎) = (√‘𝑚))
44	42, 43	oveq12d 7273	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
45		dchrisum0lem1.f	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
46		ovex 7288	. . . . . . . . . . 11 ⊢ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)) ∈ V
47	44, 45, 46	fvmpt3i 6862	. . . . . . . . . 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 677	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
50		1re 10906	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
51		elicopnf 13106	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥))
53		flge1nn 13469	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ)
54	52, 53	sylbi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ (1[,)+∞) → (⌊‘𝑥) ∈ ℕ)
55	54	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ ℕ)
56		nnuz 12550	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
57	55, 56	eleqtrdi 2849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → (⌊‘𝑥) ∈ (ℤ_≥‘1))
58	40, 23	sylanl2 677	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ)
59	49, 57, 58	fsumser 15370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
60	59	mpteq2dva 5170	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
61	39, 60	syl5eq 2791	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) = (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))))
62		fveq2 6756	. . . . . . 7 ⊢ (𝑚 = (⌊‘𝑥) → (seq1( + , 𝐹)‘𝑚) = (seq1( + , 𝐹)‘(⌊‘𝑥)))
63		rpssre 12666	. . . . . . . . 9 ⊢ ℝ⁺ ⊆ ℝ
64	63	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
65	37, 64	sstrid 3928	. . . . . . 7 ⊢ (𝜑 → (1[,)+∞) ⊆ ℝ)
66		1zzd 12281	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
67	44	cbvmptv 5183	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
68	45, 67	eqtri 2766	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)))
69	22, 68	fmptd 6970	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
70	69	ffvelrnda 6943	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ)
71	56, 66, 70	serf 13679	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
72	71	feqmptd 6819	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) = (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)))
73		dchrisum0.s	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
74	72, 73	eqbrtrrd 5094	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (seq1( + , 𝐹)‘𝑚)) ⇝ 𝑆)
75	71	ffvelrnda 6943	. . . . . . 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 15184	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (1[,)+∞) ↦ (seq1( + , 𝐹)‘(⌊‘𝑥))) ⇝_𝑟 𝑆)
79		rlimo1 15254	. . . . . 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 2839	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ↾ (1[,)+∞)) ∈ 𝑂(1))
82	24	fmpttd 6971	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))):ℝ⁺⟶ℂ)
83		1red 10907	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
84	82, 64, 83	o1resb 15203	. . . 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 256	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ 𝑂(1))
86		o1const 15257	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 𝑈 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
87	63, 27, 86	sylancr 586	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ 𝑈) ∈ 𝑂(1))
88	24, 28, 85, 87	o1mul2 15262	. 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 12282	. . . . . . . . 9 ⊢ 2 ∈ ℤ
91		rpexpcl 13729	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
92	89, 90, 91	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
93	3	nnrpd 12699	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℝ⁺)
94		rpdivcl 12684	. . . . . . . 8 ⊢ (((𝑥↑2) ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((𝑥↑2) / 𝑚) ∈ ℝ⁺)
95	92, 93, 94	syl2an 595	. . . . . . 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 26035	. . . . . . . 8 ⊢ 𝐻:ℝ⁺⟶ℝ
98	97	ffvelrni 6942	. . . . . . 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 10934	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐻‘((𝑥↑2) / 𝑚)) ∈ ℂ)
101	23, 100	mulcld 10926	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
102	1, 101	fsumcl 15373	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) ∈ ℂ)
103	24, 28	mulcld 10926	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
104	25	ad2antrr 722	. . . . . . . . 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 10926	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) ∈ ℂ)
107	1, 101, 106	fsumsub 15428	. . . . . 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 11361	. . . . . . 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 15343	. . . . . 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 15425	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · 𝑈))
111	110	oveq2d 7271	. . . . . 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 2788	. . . . 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 5170	. . . 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 11262	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈) ∈ ℂ)
115	23, 114	mulcld 10926	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
116	1, 115	fsumcl 15373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℂ)
117	116	abscld 15076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
118	115	abscld 15076	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
119	1, 118	fsumrecl 15374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ ℝ)
120		1red 10907	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℝ)
121	1, 115	fsumabs 15441	. . . . . . 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 12674	. . . . . . . . . . . 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 13444	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
126	124, 125	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
127	126, 89	rerpdivcld 12732	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ∈ ℝ)
128		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
129	128	rprecred 12712	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) ∈ ℝ)
130	23	abscld 15076	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ∈ ℝ)
131	93	rpsqrtcld 15051	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → (√‘𝑚) ∈ ℝ⁺)
132	131	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℝ⁺)
133	132	rprecred 12712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘𝑚)) ∈ ℝ)
134	114	abscld 15076	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ∈ ℝ)
135	132, 128	rpdivcld 12718	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ⁺)
136	63, 135	sselid 3915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℝ)
137	23	absge0d 15084	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))))
138	114	absge0d 15084	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)))
139	2, 3, 16	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
140	132	rpcnd 12703	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ∈ ℂ)
141	132	rpne0d 12706	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘𝑚) ≠ 0)
142	139, 140, 141	absdivd 15095	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))))
143	132	rprege0d 12708	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℝ ∧ 0 ≤ (√‘𝑚)))
144		absid 14936	. . . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (abs‘(√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
147	142, 146	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) = ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)))
148	139	abscld 15076	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
149		1red 10907	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
150		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑍) = (Base‘𝑍)
151	12	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
152		rpvmasum.a	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ)
153	152	nnnn0d 12223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
154	5, 150, 7	znzrhfo 20667	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ₀ → 𝐿:ℤ–onto→(Base‘𝑍))
155		fof 6672	. . . . . . . . . . . . . . . . . 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 13185	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (1...(⌊‘𝑥)) → 𝑚 ∈ ℤ)
159		ffvelrn 6941	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑚 ∈ ℤ) → (𝐿‘𝑚) ∈ (Base‘𝑍))
160	157, 158, 159	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝐿‘𝑚) ∈ (Base‘𝑍))
161	4, 6, 5, 150, 151, 160	dchrabs2 26315	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑋‘(𝐿‘𝑚))) ≤ 1)
162	148, 149, 132, 161	lediv1dd 12759	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((abs‘(𝑋‘(𝐿‘𝑚))) / (√‘𝑚)) ≤ (1 / (√‘𝑚)))
163	147, 162	eqbrtrd 5092	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑋‘(𝐿‘𝑚)) / (√‘𝑚))) ≤ (1 / (√‘𝑚)))
164	96, 104	divsqrtsum2 26037	. . . . . . . . . . . . . 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 683	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (abs‘((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈)) ≤ (1 / (√‘((𝑥↑2) / 𝑚))))
166	92	rprege0d 12708	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)))
167		sqrtdiv 14905	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥↑2) ∈ ℝ ∧ 0 ≤ (𝑥↑2)) ∧ 𝑚 ∈ ℝ⁺) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
168	166, 93, 167	syl2an 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚)))
169	122	ad2antlr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
170		sqrtsq 14909	. . . . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚)))
173	168, 172	eqtrd 2778	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚)))
174	173	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = (1 / (𝑥 / (√‘𝑚))))
175		rpcnne0 12677	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
176	175	ad2antlr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
177	132	rpcnne0d 12710	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
178		recdiv 11611	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
179	176, 177, 178	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (𝑥 / (√‘𝑚))) = ((√‘𝑚) / 𝑥))
180	174, 179	eqtrd 2778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / (√‘((𝑥↑2) / 𝑚))) = ((√‘𝑚) / 𝑥))
181	165, 180	breqtrd 5096	. . . . . . . . . . . 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 11847	. . . . . . . . . . 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 15094	. . . . . . . . . . 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 10901	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
185		dmdcan 11615	. . . . . . . . . . . . 13 ⊢ ((((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
186	177, 176, 184, 185	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = (1 / 𝑥))
187	135	rpcnd 12703	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → ((√‘𝑚) / 𝑥) ∈ ℂ)
188		reccl 11570	. . . . . . . . . . . . . 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 10927	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (((√‘𝑚) / 𝑥) · (1 / (√‘𝑚))) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
191	186, 190	eqtr3d 2780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘𝑥))) → (1 / 𝑥) = ((1 / (√‘𝑚)) · ((√‘𝑚) / 𝑥)))
192	182, 183, 191	3brtr4d 5102	. . . . . . . . . 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 15439	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥))
194		flge0nn0 13468	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
195		hashfz1 13988	. . . . . . . . . . . 12 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
196	123, 194, 195	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
197	196	oveq1d 7270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)) = ((⌊‘𝑥) · (1 / 𝑥)))
198	89	rpreccld 12711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
199	198	rpcnd 12703	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
200		fsumconst 15430	. . . . . . . . . . 11 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ (1 / 𝑥) ∈ ℂ) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
201	1, 199, 200	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (1 / 𝑥)))
202	126	recnd 10934	. . . . . . . . . . 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 11684	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) = ((⌊‘𝑥) · (1 / 𝑥)))
207	197, 201, 206	3eqtr4d 2788	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(1 / 𝑥) = ((⌊‘𝑥) / 𝑥))
208	193, 207	breqtrd 5096	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ ((⌊‘𝑥) / 𝑥))
209		flle 13447	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
210	124, 209	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
211	124	recnd 10934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
212	211	mulid1d 10923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 1) = 𝑥)
213	210, 212	breqtrrd 5098	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ (𝑥 · 1))
214		rpregt0 12673	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
215	214	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
216		ledivmul 11781	. . . . . . . . . 10 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
217	126, 120, 215, 216	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((⌊‘𝑥) / 𝑥) ≤ 1 ↔ (⌊‘𝑥) ≤ (𝑥 · 1)))
218	213, 217	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) / 𝑥) ≤ 1)
219	119, 127, 120, 208, 218	letrd 11062	. . . . . . 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 11062	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ≤ 1)
221	220	adantrr 713	. . . . 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 15173	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((𝐻‘((𝑥↑2) / 𝑚)) − 𝑈))) ∈ 𝑂(1))
223	113, 222	eqeltrrd 2840	. . 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 15267	. 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 256	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 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 {crab 3067 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ↾ cres 5582 ⟶wf 6414 –onto→wfo 6416 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 +∞cpnf 10937 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 (,)cioo 13008 [,)cico 13010 ...cfz 13168 ⌊cfl 13438 seqcseq 13649 ↑cexp 13710 ♯chash 13972 √csqrt 14872 abscabs 14873 ⇝ cli 15121 ⇝_𝑟 crli 15122 𝑂(1)co1 15123 Σcsu 15325 Basecbs 16840 0_gc0g 17067 ℤRHomczrh 20613 ℤ/nℤczn 20616 DChrcdchr 26285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-ec 8458 df-qs 8462 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-o1 15127 df-lo1 15128 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-dvds 15892 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-qus 17137 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-nsg 18668 df-eqg 18669 df-ghm 18747 df-cntz 18838 df-od 19051 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-sra 20349 df-rgmod 20350 df-lidl 20351 df-rsp 20352 df-2idl 20416 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-zring 20583 df-zrh 20617 df-zn 20620 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-log 25617 df-cxp 25618 df-dchr 26286

This theorem is referenced by: dchrisum0lem2 26571

W3C validator