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Theorem dchrisum0lem3 27439

Description: Lemma for dchrisum0 27440. (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)))

**Assertion**
Ref	Expression
dchrisum0lem3	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 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 𝑥,𝑊 𝑚,𝑍,𝑥,𝑦 𝐷,𝑚,𝑥,𝑦 𝐿,𝑎,𝑑,𝑚,𝑥,𝑦 𝑋,𝑎,𝑑,𝑚,𝑥,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎,𝑑) 𝑆(𝑎) 1 (𝑎,𝑑) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑚,𝑎,𝑑) 𝑁(𝑎,𝑑) 𝑊(𝑦,𝑚,𝑎,𝑑) 𝑍(𝑎,𝑑)

Proof of Theorem dchrisum0lem3 Dummy variables 𝑐 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 11237	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		sumex 15658	. . . 4 ⊢ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ V
3	2	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ V)
4		sumex 15658	. . . 4 ⊢ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ V
5	4	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ V)
6		rpvmasum.z	. . . . 5 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
7		rpvmasum.l	. . . . 5 ⊢ 𝐿 = (ℤRHom‘𝑍)
8		rpvmasum.a	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		rpvmasum2.g	. . . . 5 ⊢ 𝐺 = (DChr‘𝑁)
10		rpvmasum2.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐺)
11		rpvmasum2.1	. . . . 5 ⊢ 1 = (0_g‘𝐺)
12		rpvmasum2.w	. . . . . . . 8 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
13	12	ssrab3 4076	. . . . . . 7 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
14		difss 4127	. . . . . . 7 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
15	13, 14	sstri 3987	. . . . . 6 ⊢ 𝑊 ⊆ 𝐷
16		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
17	15, 16	sselid 3976	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
18	13, 16	sselid 3976	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
19		eldifsni 4789	. . . . . 6 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
20	18, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 1 )
21		eqid 2727	. . . . 5 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
22	6, 7, 8, 9, 10, 11, 17, 20, 21	dchrmusumlema 27413	. . . 4 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
23	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ)
24	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝑊)
25		dchrisum0lem1.f	. . . . . . 7 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎)))
26		dchrisum0.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
27	26	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝐶 ∈ (0[,)+∞))
28		dchrisum0.s	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , 𝐹) ⇝ 𝑆)
30		dchrisum0.1	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
31	30	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦)))
32		eqid 2727	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
33	32	divsqrsum 26901	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ∈ dom ⇝_𝑟
34	32	divsqrsumf 26900	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℝ
35		ax-resscn 11187	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		fss 6733	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℂ)
37	34, 35, 36	mp2an 691	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℂ
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℂ)
39		rpsup 13855	. . . . . . . . . . 11 ⊢ sup(ℝ⁺, ℝ^*, < ) = +∞
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → sup(ℝ⁺, ℝ^*, < ) = +∞)
41	38, 40	rlimdm 15519	. . . . . . . . 9 ⊢ (𝜑 → ((𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ∈ dom ⇝_𝑟 ↔ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ⇝_𝑟 ( ⇝_𝑟 ‘(𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))))))
42	33, 41	mpbii 232	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ⇝_𝑟 ( ⇝_𝑟 ‘(𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))))
43	42	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ⇝_𝑟 ( ⇝_𝑟 ‘(𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))))
44		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞))
45		simprrl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
46		simprrr 781	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))
47	6, 7, 23, 9, 10, 11, 12, 24, 25, 27, 29, 31, 32, 43, 21, 44, 45, 46	dchrisum0lem2 27438	. . . . . 6 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
48	47	rexlimdvaa 3151	. . . . 5 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1)))
49	48	exlimdv 1929	. . . 4 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1)))
50	22, 49	mpd 15	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
51	6, 7, 8, 9, 10, 11, 12, 16, 25, 26, 28, 30	dchrisum0lem1 27436	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1))
52	3, 5, 50, 51	o1add2 15592	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))) ∈ 𝑂(1))
53		ovexd 7449	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ V)
54		fzfid 13962	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
55		fzfid 13962	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
56	17	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑋 ∈ 𝐷)
57		elfzelz 13525	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥↑2))) → 𝑚 ∈ ℤ)
58	57	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℤ)
59	9, 6, 10, 7, 56, 58	dchrzrhcl 27165	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
60	59	adantr 480	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
61		elfznn 13554	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥↑2))) → 𝑚 ∈ ℕ)
62	61	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℕ)
63	62	nnrpd 13038	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℝ⁺)
64		elfznn 13554	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
65	64	nnrpd 13038	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℝ⁺)
66		rpmulcl 13021	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑚 · 𝑑) ∈ ℝ⁺)
67	63, 65, 66	syl2an 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑚 · 𝑑) ∈ ℝ⁺)
68	67	rpsqrtcld 15382	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℝ⁺)
69	68	rpcnd 13042	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℂ)
70	68	rpne0d 13045	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ≠ 0)
71	60, 69, 70	divcld 12012	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
72	55, 71	fsumcl 15703	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
73	54, 72	fsumcl 15703	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
74	73	abscld 15407	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ ℝ)
75	74	adantrr 716	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ ℝ)
76	62	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 13038	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 13047	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79	64	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ)
80	79	nnrpd 13038	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
81	80	rprege0d 13047	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
82		sqrtmul 15230	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑)) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
83	78, 81, 82	syl2anc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
84	83	oveq2d 7430	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑑))))
85	77	rpsqrtcld 15382	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑚) ∈ ℝ⁺)
86	85	rpcnne0d 13049	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
87	80	rpsqrtcld 15382	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
88	87	rpcnne0d 13049	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
89		divdiv1 11947	. . . . . . . . . 10 ⊢ (((𝑋‘(𝐿‘𝑚)) ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0) ∧ ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0)) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑑))))
90	60, 86, 88, 89	syl3anc 1369	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑑))))
91	84, 90	eqtr4d 2770	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
92	91	sumeq2dv 15673	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
93	92	sumeq2dv 15673	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
94	93	adantrr 716	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))
95		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
96	95	rpred 13040	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
97		reflcl 13785	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
98	96, 97	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
99	98	ltp1d 12166	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) < ((⌊‘𝑥) + 1))
100		fzdisj 13552	. . . . . . . 8 ⊢ ((⌊‘𝑥) < ((⌊‘𝑥) + 1) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
101	99, 100	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
102	101	adantrr 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
103	95	rprege0d 13047	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
104		flge0nn0 13809	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
105		nn0p1nn 12533	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
106	103, 104, 105	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
107		nnuz 12887	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
108	106, 107	eleqtrdi 2838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
109	108	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
110	96	adantrr 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
111		2z 12616	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
112		rpexpcl 14069	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
113	95, 111, 112	sylancl 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
114	113	adantrr 716	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ⁺)
115	114	rpred 13040	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ)
116	110	recnd 11264	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
117	116	mulridd 11253	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
118		simprr 772	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
119		1red 11237	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
120		rpregt0 13012	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
121	120	ad2antrl 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
122		lemul2 12089	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
123	119, 110, 121, 122	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
124	118, 123	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) ≤ (𝑥 · 𝑥))
125	117, 124	eqbrtrrd 5166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥 · 𝑥))
126	116	sqvald 14131	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) = (𝑥 · 𝑥))
127	125, 126	breqtrrd 5170	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥↑2))
128		flword2 13802	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ 𝑥 ≤ (𝑥↑2)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
129	110, 115, 127, 128	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
130		fzsplit2 13550	. . . . . . 7 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
131	109, 129, 130	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
132		fzfid 13962	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
133	92, 72	eqeltrrd 2829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
134	133	adantlrr 720	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
135	102, 131, 132, 134	fsumsplit 15711	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
136	94, 135	eqtrd 2767	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))))
137	136	fveq2d 6895	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))))
138	75, 137	eqled 11339	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ≤ (abs‘(Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) + Σ𝑚 ∈ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)))))
139	1, 52, 53, 73, 138	o1le 15623	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 Vcvv 3469 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 {csn 4624 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5672 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 supcsup 9455 ℂcc 11128 ℝcr 11129 0cc0 11130 1c1 11131 + caddc 11133 · cmul 11135 +∞cpnf 11267 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11466 / cdiv 11893 ℕcn 12234 2c2 12289 ℕ₀cn0 12494 ℤcz 12580 ℤ_≥cuz 12844 ℝ⁺crp 12998 [,)cico 13350 ...cfz 13508 ⌊cfl 13779 seqcseq 13990 ↑cexp 14050 √csqrt 15204 abscabs 15205 ⇝ cli 15452 ⇝_𝑟 crli 15453 𝑂(1)co1 15454 Σcsu 15656 Basecbs 17171 0_gc0g 17412 ℤRHomczrh 21412 ℤ/nℤczn 21415 DChrcdchr 27152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-fi 9426 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-acn 9957 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-xnn0 12567 df-z 12581 df-dec 12700 df-uz 12845 df-q 12955 df-rp 12999 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13352 df-ioc 13353 df-ico 13354 df-icc 13355 df-fz 13509 df-fzo 13652 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-fac 14257 df-bc 14286 df-hash 14314 df-shft 15038 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-o1 15458 df-lo1 15459 df-sum 15657 df-ef 16035 df-sin 16037 df-cos 16038 df-pi 16040 df-dvds 16223 df-gcd 16461 df-phi 16726 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-hom 17248 df-cco 17249 df-rest 17395 df-topn 17396 df-0g 17414 df-gsum 17415 df-topgen 17416 df-pt 17417 df-prds 17420 df-xrs 17475 df-qtop 17480 df-imas 17481 df-qus 17482 df-xps 17483 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-nsg 19070 df-eqg 19071 df-ghm 19159 df-cntz 19259 df-od 19474 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rsp 21094 df-2idl 21133 df-psmet 21258 df-xmet 21259 df-met 21260 df-bl 21261 df-mopn 21262 df-fbas 21263 df-fg 21264 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-zn 21419 df-top 22783 df-topon 22800 df-topsp 22822 df-bases 22836 df-cld 22910 df-ntr 22911 df-cls 22912 df-nei 22989 df-lp 23027 df-perf 23028 df-cn 23118 df-cnp 23119 df-haus 23206 df-cmp 23278 df-tx 23453 df-hmeo 23646 df-fil 23737 df-fm 23829 df-flim 23830 df-flf 23831 df-xms 24213 df-ms 24214 df-tms 24215 df-cncf 24785 df-limc 25782 df-dv 25783 df-log 26477 df-cxp 26478 df-dchr 27153

This theorem is referenced by: dchrisum0 27440

W3C validator