prmreclem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > prmreclem4		Structured version Visualization version GIF version

Theorem prmreclem4 16897

Description: Lemma for prmrec 16900. 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 16752, 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 7398	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝐾))
2	1	iuneq1d 4986	. . . . . 6 ⊢ (𝑥 = 𝐾 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘))
3	2	fveq2d 6865	. . . . 5 ⊢ (𝑥 = 𝐾 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)))
4	1	sumeq1d 15673	. . . . . 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 7406	. . . . 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 5123	. . . 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 7398	. . . . . . 7 ⊢ (𝑥 = 𝑗 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑗))
9	8	iuneq1d 4986	. . . . . 6 ⊢ (𝑥 = 𝑗 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘))
10	9	fveq2d 6865	. . . . 5 ⊢ (𝑥 = 𝑗 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)))
11	8	sumeq1d 15673	. . . . . 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 7406	. . . . 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 5123	. . . 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 7398	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...(𝑗 + 1)))
16	15	iuneq1d 4986	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘))
17	16	fveq2d 6865	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)))
18	15	sumeq1d 15673	. . . . . 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 7406	. . . . 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 5123	. . . 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 7398	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑁))
23	22	iuneq1d 4986	. . . . . 6 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
24	23	fveq2d 6865	. . . . 5 ⊢ (𝑥 = 𝑁 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
25	22	sumeq1d 15673	. . . . . 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 7406	. . . . 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 5123	. . . 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 12294	. . . . 5 ⊢ 0 ≤ 0
30		prmrec.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
31	30	nncnd 12209	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
32	31	mul01d 11380	. . . . 5 ⊢ (𝜑 → (𝑁 · 0) = 0)
33	29, 32	breqtrrid 5148	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 · 0))
34		prmrec.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℕ)
35	34	nnred 12208	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
36	35	ltp1d 12120	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 < (𝐾 + 1))
37	34	nnzd 12563	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
38	37	peano2zd 12648	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
39		fzn 13508	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 + 1)...𝐾) = ∅)
42	41	iuneq1d 4986	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∪ 𝑘 ∈ ∅ (𝑊‘𝑘))
43		0iun 5030	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ (𝑊‘𝑘) = ∅
44	42, 43	eqtrdi 2781	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∅)
45	44	fveq2d 6865	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = (♯‘∅))
46		hash0 14339	. . . . 5 ⊢ (♯‘∅) = 0
47	45, 46	eqtrdi 2781	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = 0)
48	41	sumeq1d 15673	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
49		sum0 15694	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0
50	48, 49	eqtrdi 2781	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0)
51	50	oveq2d 7406	. . . 4 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · 0))
52	33, 47, 51	3brtr4d 5142	. . 3 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
53		fzfi 13944	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ Fin
54		elfzuz 13488	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑗) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55	34	peano2nnd 12210	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ)
56		eluznn 12884	. . . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ))
59		breq1 5113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛))
60	58, 59	anbi12d 632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = 𝑘 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)))
61	60	rabbidv 3416	. . . . . . . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ∈ V
64	63	rabex 5297	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ∈ V
65	61, 62, 64	fvmpt 6971	. . . . . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . . . . 17 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ⊆ (1...𝑁)
68	66, 67	eqsstrdi 3994	. . . . . . . . . . . . . . . 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 3126	. . . . . . . . . . . . 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 5012	. . . . . . . . . . . 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 9143	. . . . . . . . . . 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 14328	. . . . . . . . . 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 12511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℝ)
80	30	nnred 12208	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
81	80	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℝ)
82		fzfid 13945	. . . . . . . . . 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 596	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → 𝑘 ∈ ℕ)
85		nnrecre 12235	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
86		0re 11183	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
87		ifcl 4537	. . . . . . . . . . . 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 15707	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
91	81, 90	remulcld 11211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
92		prmnn 16651	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℙ → (𝑗 + 1) ∈ ℕ)
93	92	nnrecred 12244	. . . . . . . . . . 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 11184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → 0 ∈ ℝ)
96	94, 95	ifclda 4527	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℝ)
97	81, 96	remulcld 11211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) ∈ ℝ)
98	79, 91, 97	leadd1d 11779	. . . . . . 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 12828	. . . . . . . . . . . . 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 13488	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
103	88	recnd 11209	. . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
107		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (1 / 𝑘) = (1 / (𝑗 + 1)))
108	106, 107	ifbieq1d 4516	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑗 + 1) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))
109	100, 105, 108	fsumm1 15724	. . . . . . . . . . 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 12810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → 𝑗 ∈ ℤ)
111	110	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℤ)
112	111	zcnd 12646	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℂ)
113		ax-1cn 11133	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		pncan 11434	. . . . . . . . . . . . . . 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 7406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...((𝑗 + 1) − 1)) = ((𝐾 + 1)...𝑗))
117	116	sumeq1d 15673	. . . . . . . . . . . 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 7405	. . . . . . . . . . 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 2765	. . . . . . . . . 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 7406	. . . . . . . . 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 11209	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
123	96	recnd 11209	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℂ)
124	121, 122, 123	adddid 11205	. . . . . . . . 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 2765	. . . . . . . 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 5122	. . . . . . 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 593	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
129	128	ralrimiva 3126	. . . . . . . . . . . . 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 5012	. . . . . . . . . . . 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 9143	. . . . . . . . . . 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 14328	. . . . . . . . . 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 12511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℝ)
138		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑗 + 1) → (𝑊‘𝑘) = (𝑊‘(𝑗 + 1)))
139	138	sseq1d 3981	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → ((𝑊‘𝑘) ⊆ (1...𝑁) ↔ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)))
140	68	ralrimiva 3126	. . . . . . . . . . . . . 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 12884	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
143	34, 142	sylan 580	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
144	143	peano2nnd 12210	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℕ)
145	139, 141, 144	rspcdva 3592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁))
146		ssfi 9143	. . . . . . . . . . . 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 14328	. . . . . . . . . . 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 12511	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℝ)
151	79, 150	readdcld 11210	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))) ∈ ℝ)
152	79, 97	readdcld 11210	. . . . . . . 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 12209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℂ)
156	155	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ ℂ)
157		pncan 11434	. . . . . . . . . . . . . . . 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 6865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘((𝐾 + 1) − 1)) = (ℤ_≥‘𝐾))
160	154, 159	eleqtrrd 2832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1)))
161		fzsuc2 13550	. . . . . . . . . . . . 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 4986	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘))
164		iunxun 5061	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘))
165		ovex 7423	. . . . . . . . . . . . . 14 ⊢ (𝑗 + 1) ∈ V
166	165, 138	iunxsn 5058	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘) = (𝑊‘(𝑗 + 1))
167	166	uneq2i 4131	. . . . . . . . . . . 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 2753	. . . . . . . . . . 11 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))
169	163, 168	eqtrdi 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1))))
170	169	fveq2d 6865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) = (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))))
171		hashun2 14355	. . . . . . . . . 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 5132	. . . . . . . 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 12247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) ∈ ℝ)
175		flle 13768	. . . . . . . . . . . . . 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 13521	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
178	177	nncnd 12209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
179	178	subid1d 11529	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 0) = 𝑛)
180	179	breq2d 5122	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑗 + 1) ∥ (𝑛 − 0) ↔ (𝑗 + 1) ∥ 𝑛))
181	180	rabbiia 3412	. . . . . . . . . . . . . . 15 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)} = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}
182	181	fveq2i 6864	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
183		1zzd 12571	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 1 ∈ ℤ)
184	30	nnnn0d 12510	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
185		nn0uz 12842	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
186		1m1e0 12265	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
187	186	fveq2i 6864	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
188	185, 187	eqtr4i 2756	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
189	184, 188	eleqtrdi 2839	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
190	189	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
191		0zd 12548	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 0 ∈ ℤ)
192	144, 183, 190, 191	hashdvds 16752	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))))
193	121	subid1d 11529	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 − 0) = 𝑁)
194	193	fvoveq1d 7412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘((𝑁 − 0) / (𝑗 + 1))) = (⌊‘(𝑁 / (𝑗 + 1))))
195	186	oveq1i 7400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − 1) − 0) = (0 − 0)
196		0m0e0 12308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 − 0) = 0
197	195, 196	eqtri 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − 1) − 0) = 0
198	197	oveq1i 7400	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 − 1) − 0) / (𝑗 + 1)) = (0 / (𝑗 + 1))
199	144	nncnd 12209	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℂ)
200	144	nnne0d 12243	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ≠ 0)
201	199, 200	div0d 11964	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (0 / (𝑗 + 1)) = 0)
202	198, 201	eqtrid 2777	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((1 − 1) − 0) / (𝑗 + 1)) = 0)
203	202	fveq2d 6865	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = (⌊‘0))
204		0z 12547	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
205		flid 13777	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (⌊‘0) = 0)
206	204, 205	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (⌊‘0) = 0
207	203, 206	eqtrdi 2781	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = 0)
208	194, 207	oveq12d 7408	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))) = ((⌊‘(𝑁 / (𝑗 + 1))) − 0))
209	174	flcld 13767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℤ)
210	209	zcnd 12646	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℂ)
211	210	subid1d 11529	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘(𝑁 / (𝑗 + 1))) − 0) = (⌊‘(𝑁 / (𝑗 + 1))))
212	192, 208, 211	3eqtrd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (⌊‘(𝑁 / (𝑗 + 1))))
213	182, 212	eqtr3id 2779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) = (⌊‘(𝑁 / (𝑗 + 1))))
214	121, 199, 200	divrecd 11968	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) = (𝑁 · (1 / (𝑗 + 1))))
215	214	eqcomd 2736	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · (1 / (𝑗 + 1))) = (𝑁 / (𝑗 + 1)))
216	176, 213, 215	3brtr4d 5142	. . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
219		breq1 5113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∥ 𝑛 ↔ (𝑗 + 1) ∥ 𝑛))
220	218, 219	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑗 + 1) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
221	220	rabbidv 3416	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑗 + 1) → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
222	63	rabex 5297	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} ∈ V
223	221, 62, 222	fvmpt 6971	. . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
230	229	fveq2d 6865	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}))
231		iftrue 4497	. . . . . . . . . . . . 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 7406	. . . . . . . . . . 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 5142	. . . . . . . . . 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 3130	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
239		rabeq0 4354	. . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = ∅)
242	241	fveq2d 6865	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘∅))
243	242, 46	eqtrdi 2781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = 0)
244		iffalse 4500	. . . . . . . . . . . . 13 ⊢ (¬ (𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = 0)
245	244	oveq2d 7406	. . . . . . . . . . . 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 2787	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = 0)
248	235, 243, 247	3brtr4d 5142	. . . . . . . . . 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 812	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
250	150, 97, 79, 249	leadd2dd 11800	. . . . . . . 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 11338	. . . . . . 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 13945	. . . . . . . . . 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 15707	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
256	81, 255	remulcld 11211	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
257		letr 11275	. . . . . . . 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 1373	. . . . . . 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 695	. . . . . 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 12876	. 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 1540 ∈ wcel 2109 ∀wral 3045 {crab 3408 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 ∅c0 4299 ifcif 4491 {csn 4592 ∪ ciun 4958 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 < clt 11215 ≤ cle 11216 − cmin 11412 / cdiv 11842 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ℤcz 12536 ℤ_≥cuz 12800 ...cfz 13475 ⌊cfl 13759 seqcseq 13973 ♯chash 14302 ⇝ cli 15457 Σcsu 15659 ∥ cdvds 16229 ℙcprime 16648

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-dvds 16230 df-prm 16649

This theorem is referenced by: prmreclem5 16898

W3C validator