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Theorem prmreclem4 17011

Description: Lemma for prmrec 17014. 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 16866, 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 7421	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝐾))
2	1	iuneq1d 4979	. . . . . 6 ⊢ (𝑥 = 𝐾 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘))
3	2	fveq2d 6882	. . . . 5 ⊢ (𝑥 = 𝐾 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)))
4	1	sumeq1d 15787	. . . . . 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 7429	. . . . 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 5116	. . . 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 343	. . 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 7421	. . . . . . 7 ⊢ (𝑥 = 𝑗 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑗))
9	8	iuneq1d 4979	. . . . . 6 ⊢ (𝑥 = 𝑗 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘))
10	9	fveq2d 6882	. . . . 5 ⊢ (𝑥 = 𝑗 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)))
11	8	sumeq1d 15787	. . . . . 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 7429	. . . . 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 5116	. . . 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 343	. . 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 7421	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...(𝑗 + 1)))
16	15	iuneq1d 4979	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘))
17	16	fveq2d 6882	. . . . 5 ⊢ (𝑥 = (𝑗 + 1) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)))
18	15	sumeq1d 15787	. . . . . 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 7429	. . . . 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 5116	. . . 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 343	. . 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 7421	. . . . . . 7 ⊢ (𝑥 = 𝑁 → ((𝐾 + 1)...𝑥) = ((𝐾 + 1)...𝑁))
23	22	iuneq1d 4979	. . . . . 6 ⊢ (𝑥 = 𝑁 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘) = ∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘))
24	23	fveq2d 6882	. . . . 5 ⊢ (𝑥 = 𝑁 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑥)(𝑊‘𝑘)) = (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑁)(𝑊‘𝑘)))
25	22	sumeq1d 15787	. . . . . 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 7429	. . . . 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 5116	. . . 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 343	. . 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 12366	. . . . 5 ⊢ 0 ≤ 0
30		prmrec.3	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
31	30	nncnd 12273	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ)
32	31	mul01d 11433	. . . . 5 ⊢ (𝜑 → (𝑁 · 0) = 0)
33	29, 32	breqtrrid 5143	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑁 · 0))
34		prmrec.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℕ)
35	34	nnred 12272	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℝ)
36	35	ltp1d 12169	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 < (𝐾 + 1))
37	34	nnzd 12641	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℤ)
38	37	peano2zd 12728	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 + 1) ∈ ℤ)
39		fzn 13594	. . . . . . . . . 10 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
40	38, 37, 39	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 < (𝐾 + 1) ↔ ((𝐾 + 1)...𝐾) = ∅))
41	36, 40	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((𝐾 + 1)...𝐾) = ∅)
42	41	iuneq1d 4979	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∪ 𝑘 ∈ ∅ (𝑊‘𝑘))
43		0iun 5021	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ (𝑊‘𝑘) = ∅
44	42, 43	eqtrdi 2811	. . . . . 6 ⊢ (𝜑 → ∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘) = ∅)
45	44	fveq2d 6882	. . . . 5 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = (♯‘∅))
46		hash0 14431	. . . . 5 ⊢ (♯‘∅) = 0
47	45, 46	eqtrdi 2811	. . . 4 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) = 0)
48	41	sumeq1d 15787	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0))
49		sum0 15807	. . . . . 6 ⊢ Σ𝑘 ∈ ∅ if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0
50	48, 49	eqtrdi 2811	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = 0)
51	50	oveq2d 7429	. . . 4 ⊢ (𝜑 → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) = (𝑁 · 0))
52	33, 47, 51	3brtr4d 5137	. . 3 ⊢ (𝜑 → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝐾)(𝑊‘𝑘)) ≤ (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝐾)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)))
53		fzfi 14036	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ Fin
54		elfzuz 13574	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ((𝐾 + 1)...𝑗) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
55	34	peano2nnd 12274	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐾 + 1) ∈ ℕ)
56		eluznn 12967	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
57	55, 56	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → 𝑘 ∈ ℕ)
58		eleq1 2848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ))
59		breq1 5106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑘 → (𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛))
60	58, 59	anbi12d 644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = 𝑘 → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)))
61	60	rabbidv 3419	. . . . . . . . . . . . . . . . . . 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 7446	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑁) ∈ V
64	63	rabex 5303	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)} ∈ V
65	61, 62, 64	fvmpt 6986	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℕ → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
66	65	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑊‘𝑘) = {𝑛 ∈ (1...𝑁) ∣ (𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛)})
67		ssrab2 4028	. . . . . . . . . . . . . . . . 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 603	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
70	54, 69	sylan2 605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → (𝑊‘𝑘) ⊆ (1...𝑁))
71	70	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
72	71	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
73		iunss 5003	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
74	72, 73	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁))
75		ssfi 9167	. . . . . . . . . . 11 ⊢ (((1...𝑁) ∈ Fin ∧ ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ⊆ (1...𝑁)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
76	53, 74, 75	sylancr 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin)
77		hashcl 14420	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∈ Fin → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℕ₀)
78	76, 77	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℕ₀)
79	78	nn0red 12590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) ∈ ℝ)
80	30	nnred 12272	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
81	80	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℝ)
82		fzfid 14037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...𝑗) ∈ Fin)
83	55	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℕ)
84	83, 54, 56	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → 𝑘 ∈ ℕ)
85		nnrecre 12302	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
86		0re 11234	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
87		ifcl 4528	. . . . . . . . . . . 12 ⊢ (((1 / 𝑘) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
88	85, 86, 87	sylancl 598	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
89	84, 88	syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...𝑗)) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
90	82, 89	fsumrecl 15820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
91	81, 90	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
92		prmnn 16764	. . . . . . . . . . . 12 ⊢ ((𝑗 + 1) ∈ ℙ → (𝑗 + 1) ∈ ℕ)
93	92	nnrecred 12311	. . . . . . . . . . 11 ⊢ ((𝑗 + 1) ∈ ℙ → (1 / (𝑗 + 1)) ∈ ℝ)
94	93	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (1 / (𝑗 + 1)) ∈ ℝ)
95		0red 11235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → 0 ∈ ℝ)
96	94, 95	ifclda 4518	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℝ)
97	81, 96	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) ∈ ℝ)
98	79, 91, 97	leadd1d 11832	. . . . . . 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 12915	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
100	99	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ (ℤ_≥‘(𝐾 + 1)))
101		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝜑)
102		elfzuz 13574	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1)) → 𝑘 ∈ (ℤ_≥‘(𝐾 + 1)))
103	88	recnd 11261	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
104	57, 103	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
105	101, 102, 104	syl2an 608	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
106		eleq1 2848	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (𝑘 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
107		oveq2 7421	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → (1 / 𝑘) = (1 / (𝑗 + 1)))
108	106, 107	ifbieq1d 4507	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑗 + 1) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) = if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))
109	100, 105, 108	fsumm1 15837	. . . . . . . . . . 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 12897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘𝐾) → 𝑗 ∈ ℤ)
111	110	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℤ)
112	111	zcnd 12726	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℂ)
113		ax-1cn 11182	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		pncan 11487	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑗 + 1) − 1) = 𝑗)
115	112, 113, 114	sylancl 598	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝑗 + 1) − 1) = 𝑗)
116	115	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...((𝑗 + 1) − 1)) = ((𝐾 + 1)...𝑗))
117	116	sumeq1d 15787	. . . . . . . . . . . 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 7428	. . . . . . . . . . 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 2795	. . . . . . . . . 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 7429	. . . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ ℂ)
122	90	recnd 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...𝑗)if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℂ)
123	96	recnd 11261	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) ∈ ℂ)
124	121, 122, 123	adddid 11257	. . . . . . . . 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 2795	. . . . . . . 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 5115	. . . . . . 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 285	. . . . . 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 605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → (𝑊‘𝑘) ⊆ (1...𝑁))
129	128	ralrimiva 3154	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
130	129	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
131		iunss 5003	. . . . . . . . . . . 12 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁) ↔ ∀𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
132	130, 131	sylibr 237	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ⊆ (1...𝑁))
133		ssfi 9167	. . . . . . . . . . 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 599	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin)
135		hashcl 14420	. . . . . . . . . 10 ⊢ (∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) ∈ Fin → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℕ₀)
136	134, 135	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℕ₀)
137	136	nn0red 12590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) ∈ ℝ)
138		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑗 + 1) → (𝑊‘𝑘) = (𝑊‘(𝑗 + 1)))
139	138	sseq1d 3962	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑗 + 1) → ((𝑊‘𝑘) ⊆ (1...𝑁) ↔ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)))
140	68	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
141	140	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∀𝑘 ∈ ℕ (𝑊‘𝑘) ⊆ (1...𝑁))
142		eluznn 12967	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ ℕ ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
143	34, 142	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ ℕ)
144	143	peano2nnd 12274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℕ)
145	139, 141, 144	rspcdva 3577	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁))
146		ssfi 9167	. . . . . . . . . . . 12 ⊢ (((1...𝑁) ∈ Fin ∧ (𝑊‘(𝑗 + 1)) ⊆ (1...𝑁)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
147	53, 145, 146	sylancr 599	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) ∈ Fin)
148		hashcl 14420	. . . . . . . . . . 11 ⊢ ((𝑊‘(𝑗 + 1)) ∈ Fin → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℕ₀)
149	147, 148	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℕ₀)
150	149	nn0red 12590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ∈ ℝ)
151	79, 150	readdcld 11262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (♯‘(𝑊‘(𝑗 + 1)))) ∈ ℝ)
152	79, 97	readdcld 11262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((♯‘∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘)) + (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0))) ∈ ℝ)
153	38	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝐾 + 1) ∈ ℤ)
154		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘𝐾))
155	34	nncnd 12273	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℂ)
156	155	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ ℂ)
157		pncan 11487	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 + 1) − 1) = 𝐾)
158	156, 113, 157	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1) − 1) = 𝐾)
159	158	fveq2d 6882	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (ℤ_≥‘((𝐾 + 1) − 1)) = (ℤ_≥‘𝐾))
160	154, 159	eleqtrrd 2863	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1)))
161		fzsuc2 13637	. . . . . . . . . . . . 13 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑗 ∈ (ℤ_≥‘((𝐾 + 1) − 1))) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
162	153, 160, 161	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...(𝑗 + 1)) = (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)}))
163	162	iuneq1d 4979	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘))
164		iunxun 5054	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘))
165		ovex 7446	. . . . . . . . . . . . . 14 ⊢ (𝑗 + 1) ∈ V
166	165, 138	iunxsn 5051	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ {(𝑗 + 1)} (𝑊‘𝑘) = (𝑊‘(𝑗 + 1))
167	166	uneq2i 4112	. . . . . . . . . . . 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 2783	. . . . . . . . . . 11 ⊢ ∪ 𝑘 ∈ (((𝐾 + 1)...𝑗) ∪ {(𝑗 + 1)})(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))
169	163, 168	eqtrdi 2811	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘) = (∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1))))
170	169	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘∪ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))(𝑊‘𝑘)) = (♯‘(∪ 𝑘 ∈ ((𝐾 + 1)...𝑗)(𝑊‘𝑘) ∪ (𝑊‘(𝑗 + 1)))))
171		hashun2 14447	. . . . . . . . . 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 596	. . . . . . . . 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 5127	. . . . . . . 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 12314	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) ∈ ℝ)
175		flle 13860	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (𝑗 + 1)) ∈ ℝ → (⌊‘(𝑁 / (𝑗 + 1))) ≤ (𝑁 / (𝑗 + 1)))
176	174, 175	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ≤ (𝑁 / (𝑗 + 1)))
177		elfznn 13608	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℕ)
178	177	nncnd 12273	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℂ)
179	178	subid1d 11582	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 − 0) = 𝑛)
180	179	breq2d 5115	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) → ((𝑗 + 1) ∥ (𝑛 − 0) ↔ (𝑗 + 1) ∥ 𝑛))
181	180	rabbiia 3416	. . . . . . . . . . . . . . 15 ⊢ {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)} = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}
182	181	fveq2i 6881	. . . . . . . . . . . . . 14 ⊢ (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
183		1zzd 12649	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 1 ∈ ℤ)
184	30	nnnn0d 12589	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
185		nn0uz 12925	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ₀ = (ℤ_≥‘0)
186		1m1e0 12337	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 − 1) = 0
187	186	fveq2i 6881	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤ_≥‘(1 − 1)) = (ℤ_≥‘0)
188	185, 187	eqtr4i 2786	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ₀ = (ℤ_≥‘(1 − 1))
189	184, 188	eleqtrdi 2870	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
190	189	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 𝑁 ∈ (ℤ_≥‘(1 − 1)))
191		0zd 12627	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → 0 ∈ ℤ)
192	144, 183, 190, 191	hashdvds 16866	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))))
193	121	subid1d 11582	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 − 0) = 𝑁)
194	193	fvoveq1d 7435	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘((𝑁 − 0) / (𝑗 + 1))) = (⌊‘(𝑁 / (𝑗 + 1))))
195	186	oveq1i 7423	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1 − 1) − 0) = (0 − 0)
196		0m0e0 12383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 − 0) = 0
197	195, 196	eqtri 2783	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1 − 1) − 0) = 0
198	197	oveq1i 7423	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 − 1) − 0) / (𝑗 + 1)) = (0 / (𝑗 + 1))
199	144	nncnd 12273	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ∈ ℂ)
200	144	nnne0d 12310	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑗 + 1) ≠ 0)
201	199, 200	div0d 12014	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (0 / (𝑗 + 1)) = 0)
202	198, 201	eqtrid 2807	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (((1 − 1) − 0) / (𝑗 + 1)) = 0)
203	202	fveq2d 6882	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = (⌊‘0))
204		0z 12626	. . . . . . . . . . . . . . . . . 18 ⊢ 0 ∈ ℤ
205		flid 13869	. . . . . . . . . . . . . . . . . 18 ⊢ (0 ∈ ℤ → (⌊‘0) = 0)
206	204, 205	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (⌊‘0) = 0
207	203, 206	eqtrdi 2811	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(((1 − 1) − 0) / (𝑗 + 1))) = 0)
208	194, 207	oveq12d 7431	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘((𝑁 − 0) / (𝑗 + 1))) − (⌊‘(((1 − 1) − 0) / (𝑗 + 1)))) = ((⌊‘(𝑁 / (𝑗 + 1))) − 0))
209	174	flcld 13859	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℤ)
210	209	zcnd 12726	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (⌊‘(𝑁 / (𝑗 + 1))) ∈ ℂ)
211	210	subid1d 11582	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((⌊‘(𝑁 / (𝑗 + 1))) − 0) = (⌊‘(𝑁 / (𝑗 + 1))))
212	192, 208, 211	3eqtrd 2799	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ (𝑛 − 0)}) = (⌊‘(𝑁 / (𝑗 + 1))))
213	182, 212	eqtr3id 2809	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) = (⌊‘(𝑁 / (𝑗 + 1))))
214	121, 199, 200	divrecd 12018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 / (𝑗 + 1)) = (𝑁 · (1 / (𝑗 + 1))))
215	214	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · (1 / (𝑗 + 1))) = (𝑁 / (𝑗 + 1)))
216	176, 213, 215	3brtr4d 5137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
217	216	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}) ≤ (𝑁 · (1 / (𝑗 + 1))))
218		eleq1 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∈ ℙ ↔ (𝑗 + 1) ∈ ℙ))
219		breq1 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑗 + 1) → (𝑝 ∥ 𝑛 ↔ (𝑗 + 1) ∥ 𝑛))
220	218, 219	anbi12d 644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = (𝑗 + 1) → ((𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛) ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
221	220	rabbidv 3419	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = (𝑗 + 1) → {𝑛 ∈ (1...𝑁) ∣ (𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛)} = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
222	63	rabex 5303	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} ∈ V
223	221, 62, 222	fvmpt 6986	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 + 1) ∈ ℕ → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
224	144, 223	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
225	224	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)})
226		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑗 + 1) ∈ ℙ)
227	226	biantrurd 542	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → ((𝑗 + 1) ∥ 𝑛 ↔ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)))
228	227	rabbidv 3419	. . . . . . . . . . . . 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 2798	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = {𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛})
230	229	fveq2d 6882	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘{𝑛 ∈ (1...𝑁) ∣ (𝑗 + 1) ∥ 𝑛}))
231		iftrue 4488	. . . . . . . . . . . . 13 ⊢ ((𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
232	231	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ (𝑗 + 1) ∈ ℙ) → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = (1 / (𝑗 + 1)))
233	232	oveq2d 7429	. . . . . . . . . . 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 5137	. . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛) → (𝑗 + 1) ∈ ℙ)
237	236	con3i 155	. . . . . . . . . . . . . . . 16 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
238	237	ralrimivw 3158	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑗 + 1) ∈ ℙ → ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
239		rabeq0 4338	. . . . . . . . . . . . . . 15 ⊢ ({𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅ ↔ ∀𝑛 ∈ (1...𝑁) ¬ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛))
240	238, 239	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (¬ (𝑗 + 1) ∈ ℙ → {𝑛 ∈ (1...𝑁) ∣ ((𝑗 + 1) ∈ ℙ ∧ (𝑗 + 1) ∥ 𝑛)} = ∅)
241	224, 240	sylan9eq 2815	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑊‘(𝑗 + 1)) = ∅)
242	241	fveq2d 6882	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = (♯‘∅))
243	242, 46	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (♯‘(𝑊‘(𝑗 + 1))) = 0)
244		iffalse 4491	. . . . . . . . . . . . 13 ⊢ (¬ (𝑗 + 1) ∈ ℙ → if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0) = 0)
245	244	oveq2d 7429	. . . . . . . . . . . 12 ⊢ (¬ (𝑗 + 1) ∈ ℙ → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = (𝑁 · 0))
246	32	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · 0) = 0)
247	245, 246	sylan9eqr 2817	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑗 + 1) ∈ ℙ) → (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)) = 0)
248	235, 243, 247	3brtr4d 5137	. . . . . . . . . 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 825	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (♯‘(𝑊‘(𝑗 + 1))) ≤ (𝑁 · if((𝑗 + 1) ∈ ℙ, (1 / (𝑗 + 1)), 0)))
250	150, 97, 79, 249	leadd2dd 11853	. . . . . . . 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 11391	. . . . . . 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 14037	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → ((𝐾 + 1)...(𝑗 + 1)) ∈ Fin)
253	57, 88	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘(𝐾 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
254	101, 102, 253	syl2an 608	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) ∧ 𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))) → if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
255	252, 254	fsumrecl 15820	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0) ∈ ℝ)
256	81, 255	remulcld 11263	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝐾)) → (𝑁 · Σ𝑘 ∈ ((𝐾 + 1)...(𝑗 + 1))if(𝑘 ∈ ℙ, (1 / 𝑘), 0)) ∈ ℝ)
257		letr 11328	. . . . . . . 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 1398	. . . . . . 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 708	. . . . . 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 243	. . . . 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 419	. . . 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 30	. . 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 12959	. 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 33	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3076 {crab 3412 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 ifcif 4482 {csn 4584 ∪ ciun 4951 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ‘cfv 6533 (class class class)co 7413 Fincfn 8952 ℂcc 11122 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 < clt 11267 ≤ cle 11268 − cmin 11465 / cdiv 11895 ℕcn 12257 2c2 12319 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 ...cfz 13561 ⌊cfl 13851 seqcseq 14065 ♯chash 14394 ⇝ cli 15571 Σcsu 15773 ∥ cdvds 16342 ℙcprime 16761

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-fl 13853 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774 df-dvds 16343 df-prm 16762

This theorem is used by: prmreclem5 17012

W3C validator