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Theorem dchrisum0lem3 27694

Description: Lemma for dchrisum0 27695. (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 11215	. 2 ⊢ (𝜑 → 1 ∈ ℝ)
2		sumex 15746	. . . 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 15746	. . . 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 4035	. . . . . . 7 ⊢ 𝑊 ⊆ (𝐷 ∖ { 1 })
14		difss 4089	. . . . . . 7 ⊢ (𝐷 ∖ { 1 }) ⊆ 𝐷
15	13, 14	sstri 3945	. . . . . 6 ⊢ 𝑊 ⊆ 𝐷
16		dchrisum0.b	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
17	15, 16	sselid 3934	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
18	13, 16	sselid 3934	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
19		eldifsni 4757	. . . . . 6 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 )
20	18, 19	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 1 )
21		eqid 2762	. . . . 5 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
22	6, 7, 8, 9, 10, 11, 17, 20, 21	dchrmusumlema 27668	. . . 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 485	. . . . . . 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 485	. . . . . . 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 485	. . . . . . 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 485	. . . . . . 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 485	. . . . . . 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 2762	. . . . . . 7 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) = (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦))))
33	32	divsqrsum 27157	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))) ∈ dom ⇝_𝑟
34	32	divsqrsumf 27156	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 · (√‘𝑦)))):ℝ⁺⟶ℝ
35		ax-resscn 11163	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
36		fss 6722	. . . . . . . . . . . 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 704	. . . . . . . . . . 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 13906	. . . . . . . . . . 11 ⊢ sup(ℝ⁺, ℝ^*, < ) = +∞
40	39	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → sup(ℝ⁺, ℝ^*, < ) = +∞)
41	38, 40	rlimdm 15609	. . . . . . . . 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 236	. . . . . . . 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 485	. . . . . . 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 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)))) → 𝑐 ∈ (0[,)+∞))
45		simprrl 792	. . . . . . 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 793	. . . . . . 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 27693	. . . . . 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 3166	. . . . 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 1962	. . . 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 16	. . 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 27691	. . 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 15682	. 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 7447	. 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 14016	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
55		fzfid 14016	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin)
56	17	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑋 ∈ 𝐷)
57		elfzelz 13558	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥↑2))) → 𝑚 ∈ ℤ)
58	57	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℤ)
59	9, 6, 10, 7, 56, 58	dchrzrhcl 27420	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
60	59	adantr 485	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
61		elfznn 13588	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥↑2))) → 𝑚 ∈ ℕ)
62	61	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℕ)
63	62	nnrpd 13064	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → 𝑚 ∈ ℝ⁺)
64		elfznn 13588	. . . . . . . . 9 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ)
65	64	nnrpd 13064	. . . . . . . 8 ⊢ (𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℝ⁺)
66		rpmulcl 13047	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑚 · 𝑑) ∈ ℝ⁺)
67	63, 65, 66	syl2an 607	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑚 · 𝑑) ∈ ℝ⁺)
68	67	rpsqrtcld 15470	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℝ⁺)
69	68	rpcnd 13068	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ∈ ℂ)
70	68	rpne0d 13071	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) ≠ 0)
71	60, 69, 70	divcld 11997	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
72	55, 71	fsumcl 15791	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
73	54, 72	fsumcl 15791	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑))) ∈ ℂ)
74	73	abscld 15497	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑚 ∈ (1...(⌊‘(𝑥↑2)))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))((𝑋‘(𝐿‘𝑚)) / (√‘(𝑚 · 𝑑)))) ∈ ℝ)
75	74	adantrr 729	. . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑚 ∈ ℕ)
77	76	nnrpd 13064	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑚 ∈ ℝ⁺)
78	77	rprege0d 13073	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑚 ∈ ℝ ∧ 0 ≤ 𝑚))
79	64	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ)
80	79	nnrpd 13064	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ⁺)
81	80	rprege0d 13073	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑))
82		sqrtmul 15317	. . . . . . . . . . 11 ⊢ (((𝑚 ∈ ℝ ∧ 0 ≤ 𝑚) ∧ (𝑑 ∈ ℝ ∧ 0 ≤ 𝑑)) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
83	78, 81, 82	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘(𝑚 · 𝑑)) = ((√‘𝑚) · (√‘𝑑)))
84	83	oveq2d 7428	. . . . . . . . 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 15470	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑚) ∈ ℝ⁺)
86	85	rpcnne0d 13075	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0))
87	80	rpsqrtcld 15470	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℝ⁺)
88	87	rpcnne0d 13075	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) ∧ 𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧ (√‘𝑑) ≠ 0))
89		divdiv1 11932	. . . . . . . . . 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 1397	. . . . . . . . 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 2800	. . . . . . . 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 15760	. . . . . . 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 15760	. . . . . 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 729	. . . . 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 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
96	95	rpred 13066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
97		reflcl 13836	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
98	96, 97	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
99	98	ltp1d 12151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) < ((⌊‘𝑥) + 1))
100		fzdisj 13586	. . . . . . . 8 ⊢ ((⌊‘𝑥) < ((⌊‘𝑥) + 1) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
101	99, 100	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
102	101	adantrr 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((1...(⌊‘𝑥)) ∩ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))) = ∅)
103	95	rprege0d 13073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
104		flge0nn0 13860	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
105		nn0p1nn 12549	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → ((⌊‘𝑥) + 1) ∈ ℕ)
106	103, 104, 105	3syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ ℕ)
107		nnuz 12907	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
108	106, 107	eleqtrdi 2872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
109	108	adantrr 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → ((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1))
110	96	adantrr 729	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
111		2z 12632	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
112		rpexpcl 14123	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 2 ∈ ℤ) → (𝑥↑2) ∈ ℝ⁺)
113	95, 111, 112	sylancl 597	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑2) ∈ ℝ⁺)
114	113	adantrr 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ⁺)
115	114	rpred 13066	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) ∈ ℝ)
116	110	recnd 11243	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
117	116	mulridd 11232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
118		simprr 784	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
119		1red 11215	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
120		rpregt0 13037	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
121	120	ad2antrl 740	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
122		lemul2 12074	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
123	119, 110, 121, 122	syl3anc 1397	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1 ≤ 𝑥 ↔ (𝑥 · 1) ≤ (𝑥 · 𝑥)))
124	118, 123	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) ≤ (𝑥 · 𝑥))
125	117, 124	eqbrtrrd 5134	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥 · 𝑥))
126	116	sqvald 14186	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (𝑥↑2) = (𝑥 · 𝑥))
127	125, 126	breqtrrd 5138	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ (𝑥↑2))
128		flword2 13853	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ (𝑥↑2) ∈ ℝ ∧ 𝑥 ≤ (𝑥↑2)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
129	110, 115, 127, 128	syl3anc 1397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥)))
130		fzsplit2 13584	. . . . . . 7 ⊢ ((((⌊‘𝑥) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘(𝑥↑2)) ∈ (ℤ_≥‘(⌊‘𝑥))) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
131	109, 129, 130	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) = ((1...(⌊‘𝑥)) ∪ (((⌊‘𝑥) + 1)...(⌊‘(𝑥↑2)))))
132		fzfid 14016	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (1...(⌊‘(𝑥↑2))) ∈ Fin)
133	92, 72	eqeltrrd 2863	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑚 ∈ (1...(⌊‘(𝑥↑2)))) → Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ)
134	133	adantlrr 733	. . . . . 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 15799	. . . . 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 2797	. . . 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 6885	. . 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 11319	. 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 15711	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∃wex 1808 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 {crab 3415 Vcvv 3454 ∖ cdif 3901 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 {csn 4588 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5660 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 supcsup 9398 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 +∞cpnf 11246 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250 − cmin 11447 / cdiv 11877 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ℤcz 12597 ℤ_≥cuz 12868 ℝ⁺crp 13022 [,)cico 13380 ...cfz 13541 ⌊cfl 13830 seqcseq 14044 ↑cexp 14104 √csqrt 15291 abscabs 15292 ⇝ cli 15542 ⇝_𝑟 crli 15543 𝑂(1)co1 15544 Σcsu 15744 Basecbs 17275 0_gc0g 17498 ℤRHomczrh 21660 ℤ/nℤczn 21663 DChrcdchr 27407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-disj 5076 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-omul 8456 df-er 8692 df-ec 8694 df-qs 8698 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-acn 9935 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-xnn0 12584 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-o1 15548 df-lo1 15549 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-dvds 16317 df-gcd 16559 df-phi 16831 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-mhm 18847 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-mulg 19140 df-subg 19195 df-nsg 19196 df-eqg 19197 df-ghm 19290 df-cntz 19393 df-od 19604 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-cring 20324 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-dvr 20490 df-rhm 20561 df-subrng 20656 df-subrg 20680 df-drng 20840 df-lmod 20994 df-lss 21064 df-lsp 21104 df-sra 21305 df-rgmod 21306 df-lidl 21343 df-rsp 21344 df-2idl 21400 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-zring 21608 df-zrh 21664 df-zn 21667 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-cmp 23555 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733 df-dchr 27408

This theorem is used by: dchrisum0 27695

W3C validator