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Theorem prmreclem4 16620

Description: Lemma for prmrec 16623. 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 16476, 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 7283	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝐾))
2	1	iuneq1d 4951	. . . . . 6 ⊢ (𝑥 = 𝐾 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘))
3	2	fveq2d 6778	. . . . 5 ⊢ (𝑥 = 𝐾 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)))
4	1	sumeq1d 15413	. . . . . 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 7291	. . . . 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 5087	. . . 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 341	. . 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 7283	. . . . . . 7 ⊢ (𝑥 = 𝑗 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑗))
9	8	iuneq1d 4951	. . . . . 6 ⊢ (𝑥 = 𝑗 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘))
10	9	fveq2d 6778	. . . . 5 ⊢ (𝑥 = 𝑗 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)))
11	8	sumeq1d 15413	. . . . . 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 7291	. . . . 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 5087	. . . 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 341	. . 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 7283	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...(𝑗 + 1)))
16	15	iuneq1d 4951	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘))
17	16	fveq2d 6778	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)))
18	15	sumeq1d 15413	. . . . . 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 7291	. . . . 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 5087	. . . 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 341	. . 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 7283	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑁))
23	22	iuneq1d 4951	. . . . . 6 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
24	23	fveq2d 6778	. . . . 5 ⊢ (𝑥 = 𝑁 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
25	22	sumeq1d 15413	. . . . . 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 7291	. . . . 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 5087	. . . 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 341	. . 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 12074	. . . . 5 ⊢ 0 ≤ 0
30		prmrec.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
31	30	nncnd 11989	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
32	31	mul01d 11174	. . . . 5 ⊢ (𝜑 → (𝑁 · 0) = 0)
33	29, 32	breqtrrid 5112	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 · 0))
34		prmrec.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℕ)
35	34	nnred 11988	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
36	35	ltp1d 11905	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 < (𝐾 + 1))
37	34	nnzd 12425	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
38	37	peano2zd 12429	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
39		fzn 13272	. . . . . . . . . 10 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
40	38, 37, 39	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
41	36, 40	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 + 1)...𝐾) = ∅)
42	41	iuneq1d 4951	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∪ 𝑘 ∈ ∅ (𝑊‘𝑘))
43		0iun 4992	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ (𝑊‘𝑘) = ∅
44	42, 43	eqtrdi 2794	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∅)
45	44	fveq2d 6778	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = (♯‘∅))
46		hash0 14082	. . . . 5 ⊢ (♯‘∅) = 0
47	45, 46	eqtrdi 2794	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = 0)
48	41	sumeq1d 15413	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
49		sum0 15433	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0
50	48, 49	eqtrdi 2794	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0)
51	50	oveq2d 7291	. . . 4 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · 0))
52	33, 47, 51	3brtr4d 5106	. . 3 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
53		fzfi 13692	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ Fin
54		elfzuz 13252	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑗) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55	34	peano2nnd 11990	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ)
56		eluznn 12658	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
57	55, 56	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
58		eleq1 2826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ))
59		breq1 5077	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛))
60	58, 59	anbi12d 631	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = 𝑘 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)))
61	60	rabbidv 3414	. . . . . . . . . . . . . . . . . . 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 7308	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ∈ V
64	63	rabex 5256	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ∈ V
65	61, 62, 64	fvmpt 6875	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
66	65	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
67		ssrab2 4013	. . . . . . . . . . . . . . . . 17 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ⊆ (1...𝑁)
68	66, 67	eqsstrdi 3975	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) ⊆ (1...𝑁))
69	57, 68	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
70	54, 69	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → (𝑊‘𝑘) ⊆ (1...𝑁))
71	70	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
72	71	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
73		iunss 4975	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
74	72, 73	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
75		ssfi 8956	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
76	53, 74, 75	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
77		hashcl 14071	. . . . . . . . . 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 12294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℝ)
80	30	nnred 11988	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
81	80	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℝ)
82		fzfid 13693	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...𝑗) ∈ Fin)
83	55	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℕ)
84	83, 54, 56	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → 𝑘 ∈ ℕ)
85		nnrecre 12015	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
86		0re 10977	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
87		ifcl 4504	. . . . . . . . . . . 12 ⊢ (((1 / 𝑘) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
88	85, 86, 87	sylancl 586	. . . . . . . . . . 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 15446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
91	81, 90	remulcld 11005	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
92		prmnn 16379	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℙ → (𝑗 + 1) ∈ ℕ)
93	92	nnrecred 12024	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ ℙ → (1 / (𝑗 + 1)) ∈ ℝ)
94	93	adantl 482	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (1 / (𝑗 + 1)) ∈ ℝ)
95		0red 10978	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → 0 ∈ ℝ)
96	94, 95	ifclda 4494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℝ)
97	81, 96	remulcld 11005	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) ∈ ℝ)
98	79, 91, 97	leadd1d 11569	. . . . . . 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 12610	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
100	99	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
101		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝜑)
102		elfzuz 13252	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
103	88	recnd 11003	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
106		eleq1 2826	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
107		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (1 / 𝑘) = (1 / (𝑗 + 1)))
108	106, 107	ifbieq1d 4483	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑗 + 1) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))
109	100, 105, 108	fsumm1 15463	. . . . . . . . . . 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 12592	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → 𝑗 ∈ ℤ)
111	110	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℤ)
112	111	zcnd 12427	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℂ)
113		ax-1cn 10929	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		pncan 11227	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
115	112, 113, 114	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝑗 + 1) − 1) = 𝑗)
116	115	oveq2d 7291	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...((𝑗 + 1) − 1)) = ((𝐾 + 1)...𝑗))
117	116	sumeq1d 15413	. . . . . . . . . . . 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 7290	. . . . . . . . . . 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 2778	. . . . . . . . . 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 7291	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℂ)
122	90	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
123	96	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℂ)
124	121, 122, 123	adddid 10999	. . . . . . . . 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 2778	. . . . . . . 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 5086	. . . . . . 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 281	. . . . . 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 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
129	128	ralrimiva 3103	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
130	129	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
131		iunss 4975	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
132	130, 131	sylibr 233	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
133		ssfi 8956	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin)
135		hashcl 14071	. . . . . . . . . 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 12294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℝ)
138		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑗 + 1) → (𝑊‘𝑘) = (𝑊‘(𝑗 + 1)))
139	138	sseq1d 3952	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → ((𝑊‘𝑘) ⊆ (1...𝑁) ↔ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)))
140	68	ralrimiva 3103	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
141	140	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
142		eluznn 12658	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
143	34, 142	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
144	143	peano2nnd 11990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℕ)
145	139, 141, 144	rspcdva 3562	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁))
146		ssfi 8956	. . . . . . . . . . . 12 ⊢ (((1...𝑁) ∈ Fin ∧ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
147	53, 145, 146	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
148		hashcl 14071	. . . . . . . . . . 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 12294	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℝ)
151	79, 150	readdcld 11004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))) ∈ ℝ)
152	79, 97	readdcld 11004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∈ ℝ)
153	38	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℤ)
154		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘𝐾))
155	34	nncnd 11989	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℂ)
156	155	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ ℂ)
157		pncan 11227	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 + 1) − 1) = 𝐾)
158	156, 113, 157	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1) − 1) = 𝐾)
159	158	fveq2d 6778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘((𝐾 + 1) − 1)) = (ℤ_≥‘𝐾))
160	154, 159	eleqtrrd 2842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1)))
161		fzsuc2 13314	. . . . . . . . . . . . 13 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1))) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
162	153, 160, 161	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
163	162	iuneq1d 4951	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘))
164		iunxun 5023	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘))
165		ovex 7308	. . . . . . . . . . . . . 14 ⊢ (𝑗 + 1) ∈ V
166	165, 138	iunxsn 5020	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘) = (𝑊‘(𝑗 + 1))
167	166	uneq2i 4094	. . . . . . . . . . . 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 2766	. . . . . . . . . . 11 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))
169	163, 168	eqtrdi 2794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1))))
170	169	fveq2d 6778	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) = (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))))
171		hashun2 14098	. . . . . . . . . 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 584	. . . . . . . . 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 5096	. . . . . . . 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 12027	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) ∈ ℝ)
175		flle 13519	. . . . . . . . . . . . . 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 13285	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
178	177	nncnd 11989	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
179	178	subid1d 11321	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 0) = 𝑛)
180	179	breq2d 5086	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑗 + 1) ∥ (𝑛 − 0) ↔ (𝑗 + 1) ∥ 𝑛))
181	180	rabbiia 3407	. . . . . . . . . . . . . . 15 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)} = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}
182	181	fveq2i 6777	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
183		1zzd 12351	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 1 ∈ ℤ)
184	30	nnnn0d 12293	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
185		nn0uz 12620	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
186		1m1e0 12045	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
187	186	fveq2i 6777	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
188	185, 187	eqtr4i 2769	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
189	184, 188	eleqtrdi 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
190	189	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
191		0zd 12331	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 0 ∈ ℤ)
192	144, 183, 190, 191	hashdvds 16476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))))
193	121	subid1d 11321	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 − 0) = 𝑁)
194	193	fvoveq1d 7297	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘((𝑁 − 0) / (𝑗 + 1))) = (⌊‘(𝑁 / (𝑗 + 1))))
195	186	oveq1i 7285	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − 1) − 0) = (0 − 0)
196		0m0e0 12093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 − 0) = 0
197	195, 196	eqtri 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − 1) − 0) = 0
198	197	oveq1i 7285	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 − 1) − 0) / (𝑗 + 1)) = (0 / (𝑗 + 1))
199	144	nncnd 11989	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℂ)
200	144	nnne0d 12023	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ≠ 0)
201	199, 200	div0d 11750	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (0 / (𝑗 + 1)) = 0)
202	198, 201	eqtrid 2790	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((1 − 1) − 0) / (𝑗 + 1)) = 0)
203	202	fveq2d 6778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = (⌊‘0))
204		0z 12330	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
205		flid 13528	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (⌊‘0) = 0)
206	204, 205	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (⌊‘0) = 0
207	203, 206	eqtrdi 2794	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = 0)
208	194, 207	oveq12d 7293	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))) = ((⌊‘(𝑁 / (𝑗 + 1))) − 0))
209	174	flcld 13518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℤ)
210	209	zcnd 12427	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℂ)
211	210	subid1d 11321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘(𝑁 / (𝑗 + 1))) − 0) = (⌊‘(𝑁 / (𝑗 + 1))))
212	192, 208, 211	3eqtrd 2782	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (⌊‘(𝑁 / (𝑗 + 1))))
213	182, 212	eqtr3id 2792	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) = (⌊‘(𝑁 / (𝑗 + 1))))
214	121, 199, 200	divrecd 11754	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) = (𝑁 · (1 / (𝑗 + 1))))
215	214	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · (1 / (𝑗 + 1))) = (𝑁 / (𝑗 + 1)))
216	176, 213, 215	3brtr4d 5106	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
217	216	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
218		eleq1 2826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
219		breq1 5077	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∥ 𝑛 ↔ (𝑗 + 1) ∥ 𝑛))
220	218, 219	anbi12d 631	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑗 + 1) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
221	220	rabbidv 3414	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑗 + 1) → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
222	63	rabex 5256	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} ∈ V
223	221, 62, 222	fvmpt 6875	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
226		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑗 + 1) ∈ ℙ)
227	226	biantrurd 533	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → ((𝑗 + 1) ∥ 𝑛 ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
228	227	rabbidv 3414	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
230	229	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}))
231		iftrue 4465	. . . . . . . . . . . . 13 ⊢ ((𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
232	231	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
233	232	oveq2d 7291	. . . . . . . . . . 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 5106	. . . . . . . . . 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 483	. . . . . . . . . . . . . . . . 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 3104	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
239		rabeq0 4318	. . . . . . . . . . . . . . 15 ⊢ ({𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅ ↔ ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
240	238, 239	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑗 + 1) ∈ ℙ → {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅)
241	224, 240	sylan9eq 2798	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = ∅)
242	241	fveq2d 6778	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘∅))
243	242, 46	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = 0)
244		iffalse 4468	. . . . . . . . . . . . 13 ⊢ (¬ (𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = 0)
245	244	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (¬ (𝑗 + 1) ∈ ℙ → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = (𝑁 · 0))
246	32	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · 0) = 0)
247	245, 246	sylan9eqr 2800	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = 0)
248	235, 243, 247	3brtr4d 5106	. . . . . . . . . 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 810	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
250	150, 97, 79, 249	leadd2dd 11590	. . . . . . . 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 11132	. . . . . . 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 13693	. . . . . . . . . 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 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
255	252, 254	fsumrecl 15446	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
256	81, 255	remulcld 11005	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
257		letr 11069	. . . . . . . 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 1370	. . . . . . 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 692	. . . . . 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 239	. . . . 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 414	. . . 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 12650	. 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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ∖ cdif 3884 ∪ cun 3885 ⊆ wss 3887 ∅c0 4256 ifcif 4459 {csn 4561 ∪ ciun 4924 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ⌊cfl 13510 seqcseq 13721 ♯chash 14044 ⇝ cli 15193 Σcsu 15397 ∥ cdvds 15963 ℙcprime 16376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-dvds 15964 df-prm 16377

This theorem is referenced by: prmreclem5 16621

W3C validator