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Theorem prmreclem4 16859

Description: Lemma for prmrec 16862. Show by induction that the indexed (nondisjoint) union 𝑊‘𝑘 is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 16714, to show that the number of numbers in 1...𝑁 that divide 𝑘 is at most 𝑁 / 𝑘. (Contributed by Mario Carneiro, 6-Aug-2014.)

**Hypotheses**
Ref	Expression
prmrec.1	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))
prmrec.2	⊢ (𝜑 → 𝐾 ∈ ℕ)
prmrec.3	⊢ (𝜑 → 𝑁 ∈ ℕ)
prmrec.4	⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}
prmrec.5	⊢ (𝜑 → seq1( + , 𝐹) ∈ dom ⇝ )
prmrec.6	⊢ (𝜑 → Σ𝑘 ∈ (ℤ_≥‘(𝐾 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) < (1 / 2))
prmrec.7	⊢ 𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)})

**Assertion**
Ref	Expression
prmreclem4	⊢ (𝜑 → (𝑁 ∈ (ℤ_≥‘𝐾) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))

Distinct variable groups: 𝑘,𝑛,𝑝,𝐹 𝑘,𝐾,𝑛,𝑝 𝑘,𝑀,𝑛,𝑝 𝜑,𝑘,𝑛,𝑝 𝑘,𝑊 𝑘,𝑁,𝑛,𝑝 Allowed substitution hints: 𝑊(𝑛,𝑝)

Proof of Theorem prmreclem4 Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝐾))
2	1	iuneq1d 4976	. . . . . 6 ⊢ (𝑥 = 𝐾 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘))
3	2	fveq2d 6846	. . . . 5 ⊢ (𝑥 = 𝐾 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)))
4	1	sumeq1d 15635	. . . . . 6 ⊢ (𝑥 = 𝐾 → Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
5	4	oveq2d 7384	. . . . 5 ⊢ (𝑥 = 𝐾 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
6	3, 5	breq12d 5113	. . . 4 ⊢ (𝑥 = 𝐾 → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
7	6	imbi2d 340	. . 3 ⊢ (𝑥 = 𝐾 → ((𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) ↔ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
8		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝑗 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑗))
9	8	iuneq1d 4976	. . . . . 6 ⊢ (𝑥 = 𝑗 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘))
10	9	fveq2d 6846	. . . . 5 ⊢ (𝑥 = 𝑗 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)))
11	8	sumeq1d 15635	. . . . . 6 ⊢ (𝑥 = 𝑗 → Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
12	11	oveq2d 7384	. . . . 5 ⊢ (𝑥 = 𝑗 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
13	10, 12	breq12d 5113	. . . 4 ⊢ (𝑥 = 𝑗 → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
14	13	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑗 → ((𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) ↔ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
15		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...(𝑗 + 1)))
16	15	iuneq1d 4976	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘))
17	16	fveq2d 6846	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)))
18	15	sumeq1d 15635	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
19	18	oveq2d 7384	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
20	17, 19	breq12d 5113	. . . 4 ⊢ (𝑥 = (𝑗 + 1) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
21	20	imbi2d 340	. . 3 ⊢ (𝑥 = (𝑗 + 1) → ((𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) ↔ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
22		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑁))
23	22	iuneq1d 4976	. . . . . 6 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
24	23	fveq2d 6846	. . . . 5 ⊢ (𝑥 = 𝑁 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
25	22	sumeq1d 15635	. . . . . 6 ⊢ (𝑥 = 𝑁 → Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
26	25	oveq2d 7384	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
27	24, 26	breq12d 5113	. . . 4 ⊢ (𝑥 = 𝑁 → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
28	27	imbi2d 340	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑥)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) ↔ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
29		0le0 12258	. . . . 5 ⊢ 0 ≤ 0
30		prmrec.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
31	30	nncnd 12173	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
32	31	mul01d 11344	. . . . 5 ⊢ (𝜑 → (𝑁 · 0) = 0)
33	29, 32	breqtrrid 5138	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 · 0))
34		prmrec.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℕ)
35	34	nnred 12172	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
36	35	ltp1d 12084	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 < (𝐾 + 1))
37	34	nnzd 12526	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
38	37	peano2zd 12611	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
39		fzn 13468	. . . . . . . . . 10 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
40	38, 37, 39	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
41	36, 40	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 + 1)...𝐾) = ∅)
42	41	iuneq1d 4976	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∪ 𝑘 ∈ ∅ (𝑊‘𝑘))
43		0iun 5020	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ (𝑊‘𝑘) = ∅
44	42, 43	eqtrdi 2788	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∅)
45	44	fveq2d 6846	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = (♯‘∅))
46		hash0 14302	. . . . 5 ⊢ (♯‘∅) = 0
47	45, 46	eqtrdi 2788	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = 0)
48	41	sumeq1d 15635	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
49		sum0 15656	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0
50	48, 49	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0)
51	50	oveq2d 7384	. . . 4 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · 0))
52	33, 47, 51	3brtr4d 5132	. . 3 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
53		fzfi 13907	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ Fin
54		elfzuz 13448	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑗) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55	34	peano2nnd 12174	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ)
56		eluznn 12843	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
57	55, 56	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
58		eleq1 2825	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ))
59		breq1 5103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛))
60	58, 59	anbi12d 633	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = 𝑘 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)))
61	60	rabbidv 3408	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = 𝑘 → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
62		prmrec.7	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑊 = (𝑝 ∈ ℕ ↦ {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)})
63		ovex 7401	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ∈ V
64	63	rabex 5286	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ∈ V
65	61, 62, 64	fvmpt 6949	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
66	65	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
67		ssrab2 4034	. . . . . . . . . . . . . . . . 17 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ⊆ (1...𝑁)
68	66, 67	eqsstrdi 3980	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) ⊆ (1...𝑁))
69	57, 68	syldan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
70	54, 69	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → (𝑊‘𝑘) ⊆ (1...𝑁))
71	70	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
73		iunss 5002	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
74	72, 73	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
75		ssfi 9109	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
76	53, 74, 75	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
77		hashcl 14291	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℕ₀)
78	76, 77	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℕ₀)
79	78	nn0red 12475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℝ)
80	30	nnred 12172	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
81	80	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℝ)
82		fzfid 13908	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...𝑗) ∈ Fin)
83	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℕ)
84	83, 54, 56	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → 𝑘 ∈ ℕ)
85		nnrecre 12199	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
86		0re 11146	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
87		ifcl 4527	. . . . . . . . . . . 12 ⊢ (((1 / 𝑘) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
88	85, 86, 87	sylancl 587	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
89	84, 88	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
90	82, 89	fsumrecl 15669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
91	81, 90	remulcld 11174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
92		prmnn 16613	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℙ → (𝑗 + 1) ∈ ℕ)
93	92	nnrecred 12208	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ ℙ → (1 / (𝑗 + 1)) ∈ ℝ)
94	93	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (1 / (𝑗 + 1)) ∈ ℝ)
95		0red 11147	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → 0 ∈ ℝ)
96	94, 95	ifclda 4517	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℝ)
97	81, 96	remulcld 11174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) ∈ ℝ)
98	79, 91, 97	leadd1d 11743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ ((𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))))
99		eluzp1p1 12791	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
100	99	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
101		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝜑)
102		elfzuz 13448	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
103	88	recnd 11172	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
104	57, 103	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
105	101, 102, 104	syl2an 597	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
106		eleq1 2825	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
107		oveq2 7376	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (1 / 𝑘) = (1 / (𝑗 + 1)))
108	106, 107	ifbieq1d 4506	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑗 + 1) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))
109	100, 105, 108	fsumm1 15686	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = (Σ𝑘 ∈ ((𝐾 + 1)...((𝑗 + 1) − 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
110		eluzelz 12773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → 𝑗 ∈ ℤ)
111	110	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℤ)
112	111	zcnd 12609	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℂ)
113		ax-1cn 11096	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		pncan 11398	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
115	112, 113, 114	sylancl 587	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝑗 + 1) − 1) = 𝑗)
116	115	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...((𝑗 + 1) − 1)) = ((𝐾 + 1)...𝑗))
117	116	sumeq1d 15635	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...((𝑗 + 1) − 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
118	117	oveq1d 7383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (Σ𝑘 ∈ ((𝐾 + 1)...((𝑗 + 1) − 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = (Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
119	109, 118	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = (Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
120	119	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · (Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))))
121	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℂ)
122	90	recnd 11172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
123	96	recnd 11172	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℂ)
124	121, 122, 123	adddid 11168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · (Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) + if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) = ((𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))))
125	120, 124	eqtrd 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = ((𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))))
126	125	breq2d 5112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ ((𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))))
127	98, 126	bitr4d 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ↔ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
128	102, 69	sylan2 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
129	128	ralrimiva 3130	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
130	129	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
131		iunss 5002	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
132	130, 131	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
133		ssfi 9109	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin)
134	53, 132, 133	sylancr 588	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin)
135		hashcl 14291	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℕ₀)
136	134, 135	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℕ₀)
137	136	nn0red 12475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℝ)
138		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑗 + 1) → (𝑊‘𝑘) = (𝑊‘(𝑗 + 1)))
139	138	sseq1d 3967	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → ((𝑊‘𝑘) ⊆ (1...𝑁) ↔ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)))
140	68	ralrimiva 3130	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
141	140	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
142		eluznn 12843	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
143	34, 142	sylan 581	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
144	143	peano2nnd 12174	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℕ)
145	139, 141, 144	rspcdva 3579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁))
146		ssfi 9109	. . . . . . . . . . . 12 ⊢ (((1...𝑁) ∈ Fin ∧ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
147	53, 145, 146	sylancr 588	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
148		hashcl 14291	. . . . . . . . . . 11 ⊢ ((𝑊‘(𝑗 + 1)) ∈ Fin → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℕ₀)
149	147, 148	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℕ₀)
150	149	nn0red 12475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℝ)
151	79, 150	readdcld 11173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))) ∈ ℝ)
152	79, 97	readdcld 11173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∈ ℝ)
153	38	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℤ)
154		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘𝐾))
155	34	nncnd 12173	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℂ)
156	155	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ ℂ)
157		pncan 11398	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 + 1) − 1) = 𝐾)
158	156, 113, 157	sylancl 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1) − 1) = 𝐾)
159	158	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘((𝐾 + 1) − 1)) = (ℤ_≥‘𝐾))
160	154, 159	eleqtrrd 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1)))
161		fzsuc2 13510	. . . . . . . . . . . . 13 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1))) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
162	153, 160, 161	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
163	162	iuneq1d 4976	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘))
164		iunxun 5051	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘))
165		ovex 7401	. . . . . . . . . . . . . 14 ⊢ (𝑗 + 1) ∈ V
166	165, 138	iunxsn 5048	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘) = (𝑊‘(𝑗 + 1))
167	166	uneq2i 4119	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘)) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))
168	164, 167	eqtri 2760	. . . . . . . . . . 11 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))
169	163, 168	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1))))
170	169	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) = (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))))
171		hashun2 14318	. . . . . . . . . 10 ⊢ ((∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin ∧ (𝑊‘(𝑗 + 1)) ∈ Fin) → (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))))
172	76, 147, 171	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))))
173	170, 172	eqbrtrd 5122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))))
174	81, 144	nndivred 12211	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) ∈ ℝ)
175		flle 13731	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (𝑗 + 1)) ∈ ℝ → (⌊‘(𝑁 / (𝑗 + 1))) ≤ (𝑁 / (𝑗 + 1)))
176	174, 175	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ≤ (𝑁 / (𝑗 + 1)))
177		elfznn 13481	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
178	177	nncnd 12173	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
179	178	subid1d 11493	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 0) = 𝑛)
180	179	breq2d 5112	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑗 + 1) ∥ (𝑛 − 0) ↔ (𝑗 + 1) ∥ 𝑛))
181	180	rabbiia 3405	. . . . . . . . . . . . . . 15 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)} = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}
182	181	fveq2i 6845	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
183		1zzd 12534	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 1 ∈ ℤ)
184	30	nnnn0d 12474	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
185		nn0uz 12801	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
186		1m1e0 12229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
187	186	fveq2i 6845	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
188	185, 187	eqtr4i 2763	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
189	184, 188	eleqtrdi 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
190	189	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
191		0zd 12512	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 0 ∈ ℤ)
192	144, 183, 190, 191	hashdvds 16714	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))))
193	121	subid1d 11493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 − 0) = 𝑁)
194	193	fvoveq1d 7390	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘((𝑁 − 0) / (𝑗 + 1))) = (⌊‘(𝑁 / (𝑗 + 1))))
195	186	oveq1i 7378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − 1) − 0) = (0 − 0)
196		0m0e0 12272	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 − 0) = 0
197	195, 196	eqtri 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − 1) − 0) = 0
198	197	oveq1i 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 − 1) − 0) / (𝑗 + 1)) = (0 / (𝑗 + 1))
199	144	nncnd 12173	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℂ)
200	144	nnne0d 12207	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ≠ 0)
201	199, 200	div0d 11928	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (0 / (𝑗 + 1)) = 0)
202	198, 201	eqtrid 2784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((1 − 1) − 0) / (𝑗 + 1)) = 0)
203	202	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = (⌊‘0))
204		0z 12511	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
205		flid 13740	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (⌊‘0) = 0)
206	204, 205	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (⌊‘0) = 0
207	203, 206	eqtrdi 2788	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = 0)
208	194, 207	oveq12d 7386	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))) = ((⌊‘(𝑁 / (𝑗 + 1))) − 0))
209	174	flcld 13730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℤ)
210	209	zcnd 12609	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℂ)
211	210	subid1d 11493	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘(𝑁 / (𝑗 + 1))) − 0) = (⌊‘(𝑁 / (𝑗 + 1))))
212	192, 208, 211	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (⌊‘(𝑁 / (𝑗 + 1))))
213	182, 212	eqtr3id 2786	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) = (⌊‘(𝑁 / (𝑗 + 1))))
214	121, 199, 200	divrecd 11932	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) = (𝑁 · (1 / (𝑗 + 1))))
215	214	eqcomd 2743	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · (1 / (𝑗 + 1))) = (𝑁 / (𝑗 + 1)))
216	176, 213, 215	3brtr4d 5132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
217	216	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
218		eleq1 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
219		breq1 5103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∥ 𝑛 ↔ (𝑗 + 1) ∥ 𝑛))
220	218, 219	anbi12d 633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑗 + 1) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
221	220	rabbidv 3408	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑗 + 1) → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
222	63	rabex 5286	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} ∈ V
223	221, 62, 222	fvmpt 6949	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 + 1) ∈ ℕ → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
224	144, 223	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
225	224	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
226		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑗 + 1) ∈ ℙ)
227	226	biantrurd 532	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → ((𝑗 + 1) ∥ 𝑛 ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
228	227	rabbidv 3408	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛} = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
229	225, 228	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
230	229	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}))
231		iftrue 4487	. . . . . . . . . . . . 13 ⊢ ((𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
232	231	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
233	232	oveq2d 7384	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = (𝑁 · (1 / (𝑗 + 1))))
234	217, 230, 233	3brtr4d 5132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
235	29	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → 0 ≤ 0)
236		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛) → (𝑗 + 1) ∈ ℙ)
237	236	con3i 154	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
238	237	ralrimivw 3134	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
239		rabeq0 4342	. . . . . . . . . . . . . . 15 ⊢ ({𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅ ↔ ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
240	238, 239	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑗 + 1) ∈ ℙ → {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅)
241	224, 240	sylan9eq 2792	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = ∅)
242	241	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘∅))
243	242, 46	eqtrdi 2788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = 0)
244		iffalse 4490	. . . . . . . . . . . . 13 ⊢ (¬ (𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = 0)
245	244	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (¬ (𝑗 + 1) ∈ ℙ → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = (𝑁 · 0))
246	32	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · 0) = 0)
247	245, 246	sylan9eqr 2794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = 0)
248	235, 243, 247	3brtr4d 5132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
249	234, 248	pm2.61dan 813	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
250	150, 97, 79, 249	leadd2dd 11764	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))))
251	137, 151, 152, 173, 250	letrd 11302	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))))
252		fzfid 13908	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...(𝑗 + 1)) ∈ Fin)
253	57, 88	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
254	101, 102, 253	syl2an 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
255	252, 254	fsumrecl 15669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
256	81, 255	remulcld 11174	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
257		letr 11239	. . . . . . . 8 ⊢ (((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℝ ∧ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∈ ℝ ∧ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ) → (((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∧ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
258	137, 152, 256, 257	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∧ ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
259	251, 258	mpand 696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
260	127, 259	sylbid 240	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
261	260	expcom 413	. . . 4 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → (𝜑 → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
262	261	a2d 29	. . 3 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → ((𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))) → (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))))
263	7, 14, 21, 28, 52, 262	uzind4i 12835	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘𝐾) → (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))
264	263	com12 32	1 ⊢ (𝜑 → (𝑁 ∈ (ℤ_≥‘𝐾) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑁)if(𝑘 ∈ ℙ, (1 / 𝑘), 0))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 {crab 3401 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 ifcif 4481 {csn 4582 ∪ ciun 4948 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 ⌊cfl 13722 seqcseq 13936 ♯chash 14265 ⇝ cli 15419 Σcsu 15621 ∥ cdvds 16191 ℙcprime 16610

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-dvds 16192 df-prm 16611

This theorem is referenced by: prmreclem5 16860

W3C validator