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Theorem dchrisum0lem3 27577

Description: Lemma for dchrisum0 27578. (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 11259	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		sumex 15720	. . . 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 15720	. . . 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 4091	. . . . . . 7 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
14		difss 4145	. . . . . . 7 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
15	13, 14	sstri 4004	. . . . . 6 ⊢ 𝑊 ⊆ 𝐷
16		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
17	15, 16	sselid 3992	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
18	13, 16	sselid 3992	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
19		eldifsni 4794	. . . . . 6 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
20	18, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 1 )
21		eqid 2734	. . . . 5 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
22	6, 7, 8, 9, 10, 11, 17, 20, 21	dchrmusumlema 27551	. . . 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 2734	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
33	32	divsqrsum 27039	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ∈ dom ⇝_𝑟
34	32	divsqrsumf 27038	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℝ
35		ax-resscn 11209	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		fss 6752	. . . . . . . . . . . 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 692	. . . . . . . . . . 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 13902	. . . . . . . . . . 11 ⊢ sup(ℝ⁺, ℝ^*, < ) = +∞
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → sup(ℝ⁺, ℝ^*, < ) = +∞)
41	38, 40	rlimdm 15583	. . . . . . . . 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 233	. . . . . . . 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 771	. . . . . . 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 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)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
46		simprrr 782	. . . . . . 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 27576	. . . . . 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 3153	. . . . 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 1930	. . . 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 27574	. . 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 15656	. 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 7465	. 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 14010	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
55		fzfid 14010	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
56	17	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑋 ∈ 𝐷)
57		elfzelz 13560	. . . . . . . 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 27303	. . . . . 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 13589	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥↑2))) → 𝑚 ∈ ℕ)
62	61	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℕ)
63	62	nnrpd 13072	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℝ⁺)
64		elfznn 13589	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
65	64	nnrpd 13072	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℝ⁺)
66		rpmulcl 13055	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑚 · 𝑑) ∈ ℝ⁺)
67	63, 65, 66	syl2an 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑚 · 𝑑) ∈ ℝ⁺)
68	67	rpsqrtcld 15446	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℝ⁺)
69	68	rpcnd 13076	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℂ)
70	68	rpne0d 13079	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ≠ 0)
71	60, 69, 70	divcld 12040	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
72	55, 71	fsumcl 15765	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
73	54, 72	fsumcl 15765	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
74	73	abscld 15471	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ ℝ)
75	74	adantrr 717	. . 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 13072	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 13081	. . . . . . . . . . 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 13072	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
81	80	rprege0d 13081	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
82		sqrtmul 15294	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑)) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
83	78, 81, 82	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
84	83	oveq2d 7446	. . . . . . . . 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 15446	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑚) ∈ ℝ⁺)
86	85	rpcnne0d 13083	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
87	80	rpsqrtcld 15446	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
88	87	rpcnne0d 13083	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
89		divdiv1 11975	. . . . . . . . . 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 1370	. . . . . . . . 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 2777	. . . . . . . 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 15734	. . . . . . 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 15734	. . . . . 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 717	. . . . 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 13074	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
97		reflcl 13832	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
98	96, 97	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
99	98	ltp1d 12195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) < ((⌊‘𝑥) + 1))
100		fzdisj 13587	. . . . . . . 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 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
103	95	rprege0d 13081	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
104		flge0nn0 13856	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
105		nn0p1nn 12562	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
106	103, 104, 105	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
107		nnuz 12918	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
108	106, 107	eleqtrdi 2848	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
109	108	adantrr 717	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
110	96	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
111		2z 12646	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
112		rpexpcl 14117	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
113	95, 111, 112	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
114	113	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ⁺)
115	114	rpred 13074	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ)
116	110	recnd 11286	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
117	116	mulridd 11275	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
118		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
119		1red 11259	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
120		rpregt0 13046	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
121	120	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
122		lemul2 12117	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
123	119, 110, 121, 122	syl3anc 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
124	118, 123	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) ≤ (𝑥 · 𝑥))
125	117, 124	eqbrtrrd 5171	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥 · 𝑥))
126	116	sqvald 14179	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) = (𝑥 · 𝑥))
127	125, 126	breqtrrd 5175	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥↑2))
128		flword2 13849	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ 𝑥 ≤ (𝑥↑2)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
129	110, 115, 127, 128	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
130		fzsplit2 13585	. . . . . . 7 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
131	109, 129, 130	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
132		fzfid 14010	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
133	92, 72	eqeltrrd 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
134	133	adantlrr 721	. . . . . 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 15773	. . . . 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 2774	. . . 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 6910	. . 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 11361	. 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 15685	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 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 supcsup 9477 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 +∞cpnf 11289 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 ℕcn 12263 2c2 12318 ℕ₀cn0 12523 ℤcz 12610 ℤ_≥cuz 12875 ℝ⁺crp 13031 [,)cico 13385 ...cfz 13543 ⌊cfl 13826 seqcseq 14038 ↑cexp 14098 √csqrt 15268 abscabs 15269 ⇝ cli 15516 ⇝_𝑟 crli 15517 𝑂(1)co1 15518 Σcsu 15718 Basecbs 17244 0_gc0g 17485 ℤRHomczrh 21527 ℤ/nℤczn 21530 DChrcdchr 27290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-oadd 8508 df-omul 8509 df-er 8743 df-ec 8745 df-qs 8749 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-acn 9979 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-o1 15522 df-lo1 15523 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-dvds 16287 df-gcd 16528 df-phi 16799 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-nsg 19154 df-eqg 19155 df-ghm 19243 df-cntz 19347 df-od 19560 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-drng 20747 df-lmod 20876 df-lss 20947 df-lsp 20987 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-rsp 21236 df-2idl 21277 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-zring 21475 df-zrh 21531 df-zn 21534 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-cmp 23410 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-cxp 26613 df-dchr 27291

This theorem is referenced by: dchrisum0 27578

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