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Theorem prmreclem2 16845

Description: Lemma for prmrec 16850. There are at most 2↑𝐾 squarefree numbers which divide no primes larger than 𝐾. (We could strengthen this to 2↑♯(ℙ ∩ (1...𝐾)) but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to 𝐾 completely determine it because all higher prime counts are zero, and they are all at most 1 because no square divides the number, so there are at most 2↑𝐾 possibilities. (Contributed by Mario Carneiro, 5-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}
prmreclem2.5	⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))

**Assertion**
Ref	Expression
prmreclem2	⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾))

Distinct variable groups: 𝑛,𝑝,𝑟,𝑥,𝐹 𝑛,𝐾,𝑝,𝑥 𝑛,𝑀,𝑝,𝑥 𝜑,𝑛,𝑝,𝑥 𝑄,𝑛,𝑝,𝑟,𝑥 𝑛,𝑁,𝑝,𝑥 Allowed substitution hints: 𝜑(𝑟) 𝐾(𝑟) 𝑀(𝑟) 𝑁(𝑟)

Proof of Theorem prmreclem2 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7391	. . . 4 ⊢ ({0, 1} ↑_m (1...𝐾)) ∈ V
2		fveqeq2 6843	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑦) = 1))
3	2	elrab 3646	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1))
4		prmrec.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}
5	4	ssrab3 4034	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑀 ⊆ (1...𝑁)
6		simprl 770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → 𝑦 ∈ 𝑀)
7	6	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ 𝑀)
8	5, 7	sselid 3931	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ (1...𝑁))
9		elfznn 13469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
10	8, 9	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℕ)
11		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ ℙ)
12		prmuz2 16623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℙ → 𝑛 ∈ (ℤ_≥‘2))
13	11, 12	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑛 ∈ (ℤ_≥‘2))
14		prmreclem2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
15	14	prmreclem1 16844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ_≥‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))))
16	15	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → (𝑛 ∈ (ℤ_≥‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
17	10, 13, 16	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))
18		simprr 772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑄‘𝑦) = 1)
19	18	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑄‘𝑦) = 1)
20	19	oveq1d 7373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = (1↑2))
21		sq1 14118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (1↑2) = 1
22	20, 21	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑄‘𝑦)↑2) = 1)
23	22	oveq2d 7374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = (𝑦 / 1))
24	10	nncnd 12161	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℂ)
25	24	div1d 11909	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / 1) = 𝑦)
26	23, 25	eqtrd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑦 / ((𝑄‘𝑦)↑2)) = 𝑦)
27	26	breq2d 5110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ (𝑛↑2) ∥ 𝑦))
28	10	nnzd 12514	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 𝑦 ∈ ℤ)
29		2nn0 12418	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℕ₀
30	29	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → 2 ∈ ℕ₀)
31		pcdvdsb 16797	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℙ ∧ 𝑦 ∈ ℤ ∧ 2 ∈ ℕ₀) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
32	11, 28, 30, 31	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (2 ≤ (𝑛 pCnt 𝑦) ↔ (𝑛↑2) ∥ 𝑦))
33	27, 32	bitr4d 282	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ↔ 2 ≤ (𝑛 pCnt 𝑦)))
34	17, 33	mtbid 324	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ¬ 2 ≤ (𝑛 pCnt 𝑦))
35	11, 10	pccld 16778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℕ₀)
36	35	nn0red 12463	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℝ)
37		2re 12219	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
38		ltnle 11212	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 pCnt 𝑦) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
39	36, 37, 38	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) < 2 ↔ ¬ 2 ≤ (𝑛 pCnt 𝑦)))
40	34, 39	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < 2)
41		df-2 12208	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
42	40, 41	breqtrdi 5139	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) < (1 + 1))
43	35	nn0zd 12513	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ ℤ)
44		1z 12521	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
45		zleltp1 12542	. . . . . . . . . . . . . 14 ⊢ (((𝑛 pCnt 𝑦) ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
46	43, 44, 45	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ≤ 1 ↔ (𝑛 pCnt 𝑦) < (1 + 1)))
47	42, 46	mpbird 257	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ≤ 1)
48		nn0uz 12789	. . . . . . . . . . . . . 14 ⊢ ℕ₀ = (ℤ_≥‘0)
49	35, 48	eleqtrdi 2846	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (ℤ_≥‘0))
50		elfz5 13432	. . . . . . . . . . . . 13 ⊢ (((𝑛 pCnt 𝑦) ∈ (ℤ_≥‘0) ∧ 1 ∈ ℤ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
51	49, 44, 50	sylancl 586	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → ((𝑛 pCnt 𝑦) ∈ (0...1) ↔ (𝑛 pCnt 𝑦) ≤ 1))
52	47, 51	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ (0...1))
53		0z 12499	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℤ
54		fzpr 13495	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℤ → (0...(0 + 1)) = {0, (0 + 1)})
55	53, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0...(0 + 1)) = {0, (0 + 1)}
56		1e0p1 12649	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
57	56	oveq2i 7369	. . . . . . . . . . . 12 ⊢ (0...1) = (0...(0 + 1))
58	56	preq2i 4694	. . . . . . . . . . . 12 ⊢ {0, 1} = {0, (0 + 1)}
59	55, 57, 58	3eqtr4i 2769	. . . . . . . . . . 11 ⊢ (0...1) = {0, 1}
60	52, 59	eleqtrdi 2846	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ 𝑛 ∈ ℙ) → (𝑛 pCnt 𝑦) ∈ {0, 1})
61		c0ex 11126	. . . . . . . . . . . 12 ⊢ 0 ∈ V
62	61	prid1 4719	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1}
63	62	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) ∧ ¬ 𝑛 ∈ ℙ) → 0 ∈ {0, 1})
64	60, 63	ifclda 4515	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) ∧ 𝑛 ∈ (1...𝐾)) → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ {0, 1})
65	64	fmpttd 7060	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
66		prex 5382	. . . . . . . . 9 ⊢ {0, 1} ∈ V
67		ovex 7391	. . . . . . . . 9 ⊢ (1...𝐾) ∈ V
68	66, 67	elmap 8809	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑_m (1...𝐾)) ↔ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)):(1...𝐾)⟶{0, 1})
69	65, 68	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1)) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑_m (1...𝐾)))
70	69	ex 412	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑_m (1...𝐾))))
71	3, 70	biimtrid 242	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} → (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) ∈ ({0, 1} ↑_m (1...𝐾))))
72		fveqeq2 6843	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝑄‘𝑥) = 1 ↔ (𝑄‘𝑧) = 1))
73	72	elrab 3646	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))
74	3, 73	anbi12i 628	. . . . . 6 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ 𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ↔ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)))
75		ovex 7391	. . . . . . . . . . . 12 ⊢ (𝑛 pCnt 𝑦) ∈ V
76	75, 61	ifex 4530	. . . . . . . . . . 11 ⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) ∈ V
77		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))
78	76, 77	fnmpti 6635	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾)
79		ovex 7391	. . . . . . . . . . . 12 ⊢ (𝑛 pCnt 𝑧) ∈ V
80	79, 61	ifex 4530	. . . . . . . . . . 11 ⊢ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) ∈ V
81		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))
82	80, 81	fnmpti 6635	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)
83		eqfnfv 6976	. . . . . . . . . 10 ⊢ (((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) Fn (1...𝐾) ∧ (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) Fn (1...𝐾)) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝)))
84	78, 82, 83	mp2an 692	. . . . . . . . 9 ⊢ ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝))
85		eleq1w 2819	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑝 → (𝑛 ∈ ℙ ↔ 𝑝 ∈ ℙ))
86		oveq1 7365	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑦) = (𝑝 pCnt 𝑦))
87	85, 86	ifbieq1d 4504	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
88		ovex 7391	. . . . . . . . . . . . 13 ⊢ (𝑝 pCnt 𝑦) ∈ V
89	88, 61	ifex 4530	. . . . . . . . . . . 12 ⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) ∈ V
90	87, 77, 89	fvmpt 6941	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0))
91		oveq1 7365	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑝 → (𝑛 pCnt 𝑧) = (𝑝 pCnt 𝑧))
92	85, 91	ifbieq1d 4504	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑝 → if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
93		ovex 7391	. . . . . . . . . . . . 13 ⊢ (𝑝 pCnt 𝑧) ∈ V
94	93, 61	ifex 4530	. . . . . . . . . . . 12 ⊢ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∈ V
95	92, 81, 94	fvmpt 6941	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (1...𝐾) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
96	90, 95	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑝 ∈ (1...𝐾) → (((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) ↔ if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
97	96	ralbiia 3080	. . . . . . . . 9 ⊢ (∀𝑝 ∈ (1...𝐾)((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0))‘𝑝) = ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0))‘𝑝) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
98	84, 97	bitri 275	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
99		simprll 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ 𝑀)
100		breq2 5102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑦))
101	100	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑦))
102	101	ralbidv 3159	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦))
103	102, 4	elrab2 3649	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦))
104	103	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)
105	99, 104	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)
106		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ 𝑀)
107		breq2 5102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑧 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑧))
108	107	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑧 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑧))
109	108	ralbidv 3159	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑧 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧))
110	109, 4	elrab2 3649	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑀 ↔ (𝑧 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧))
111	110	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑀 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)
112	106, 111	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧)
113		r19.26 3096	. . . . . . . . . . . . 13 ⊢ (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) ↔ (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧))
114		eldifi 4083	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℙ)
115		fz1ssnn 13471	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...𝑁) ⊆ ℕ
116	5, 115	sstri 3943	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑀 ⊆ ℕ
117	116, 99	sselid 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ)
118		pceq0 16799	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ ℙ ∧ 𝑦 ∈ ℕ) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦))
119	114, 117, 118	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑦) = 0 ↔ ¬ 𝑝 ∥ 𝑦))
120	116, 106	sselid 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ)
121		pceq0 16799	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ∈ ℙ ∧ 𝑧 ∈ ℕ) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧))
122	114, 120, 121	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 pCnt 𝑧) = 0 ↔ ¬ 𝑝 ∥ 𝑧))
123	119, 122	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) ↔ (¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧)))
124		eqtr3 2758	. . . . . . . . . . . . . . 15 ⊢ (((𝑝 pCnt 𝑦) = 0 ∧ (𝑝 pCnt 𝑧) = 0) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
125	123, 124	biimtrrdi 254	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) → (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
126	125	ralimdva 3148	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾))(¬ 𝑝 ∥ 𝑦 ∧ ¬ 𝑝 ∥ 𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
127	113, 126	biimtrrid 243	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ((∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑧) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
128	105, 112, 127	mp2and 699	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
129	128	biantrud 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))))
130		incom 4161	. . . . . . . . . . . . . . 15 ⊢ (ℙ ∩ (1...𝐾)) = ((1...𝐾) ∩ ℙ)
131	130	uneq1i 4116	. . . . . . . . . . . . . 14 ⊢ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ))
132		inundif 4431	. . . . . . . . . . . . . 14 ⊢ (((1...𝐾) ∩ ℙ) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
133	131, 132	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ)) = (1...𝐾)
134	133	raleqi 3294	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
135		ralunb 4149	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ ((1...𝐾) ∖ ℙ))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
136	134, 135	bitr3i 277	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
137		eldifn 4084	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → ¬ 𝑝 ∈ ℙ)
138		iffalse 4488	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = 0)
139		iffalse 4488	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = 0)
140	138, 139	eqtr4d 2774	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
141	137, 140	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ((1...𝐾) ∖ ℙ) → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0))
142	141	rgen 3053	. . . . . . . . . . . . 13 ⊢ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)
143	142	biantru 529	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)))
144		elinel1 4153	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℙ ∩ (1...𝐾)) → 𝑝 ∈ ℙ)
145		iftrue 4485	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = (𝑝 pCnt 𝑦))
146		iftrue 4485	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ ℙ → if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) = (𝑝 pCnt 𝑧))
147	145, 146	eqeq12d 2752	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ ℙ → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
148	144, 147	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (ℙ ∩ (1...𝐾)) → (if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
149	148	ralbiia 3080	. . . . . . . . . . . 12 ⊢ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
150	143, 149	bitr3i 277	. . . . . . . . . . 11 ⊢ ((∀𝑝 ∈ (ℙ ∩ (1...𝐾))if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ∧ ∀𝑝 ∈ ((1...𝐾) ∖ ℙ)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0)) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
151	136, 150	bitri 275	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
152		inundif 4431	. . . . . . . . . . . 12 ⊢ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾))) = ℙ
153	152	raleqi 3294	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧))
154		ralunb 4149	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ ((ℙ ∩ (1...𝐾)) ∪ (ℙ ∖ (1...𝐾)))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
155	153, 154	bitr3i 277	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ↔ (∀𝑝 ∈ (ℙ ∩ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾))(𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
156	129, 151, 155	3bitr4g 314	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
157	117	nnnn0d 12462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑦 ∈ ℕ₀)
158	120	nnnn0d 12462	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → 𝑧 ∈ ℕ₀)
159		pc11 16808	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝑧 ∈ ℕ₀) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
160	157, 158, 159	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (𝑦 = 𝑧 ↔ ∀𝑝 ∈ ℙ (𝑝 pCnt 𝑦) = (𝑝 pCnt 𝑧)))
161	156, 160	bitr4d 282	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → (∀𝑝 ∈ (1...𝐾)if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑦), 0) = if(𝑝 ∈ ℙ, (𝑝 pCnt 𝑧), 0) ↔ 𝑦 = 𝑧))
162	98, 161	bitrid 283	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1))) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧))
163	162	ex 412	. . . . . 6 ⊢ (𝜑 → (((𝑦 ∈ 𝑀 ∧ (𝑄‘𝑦) = 1) ∧ (𝑧 ∈ 𝑀 ∧ (𝑄‘𝑧) = 1)) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧)))
164	74, 163	biimtrid 242	. . . . 5 ⊢ (𝜑 → ((𝑦 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ 𝑧 ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) → ((𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑦), 0)) = (𝑛 ∈ (1...𝐾) ↦ if(𝑛 ∈ ℙ, (𝑛 pCnt 𝑧), 0)) ↔ 𝑦 = 𝑧)))
165	71, 164	dom2d 8930	. . . 4 ⊢ (𝜑 → (({0, 1} ↑_m (1...𝐾)) ∈ V → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑_m (1...𝐾))))
166	1, 165	mpi 20	. . 3 ⊢ (𝜑 → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑_m (1...𝐾)))
167		fzfi 13895	. . . . . . 7 ⊢ (1...𝑁) ∈ Fin
168		ssrab2 4032	. . . . . . 7 ⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁)
169		ssfi 9097	. . . . . . 7 ⊢ (((1...𝑁) ∈ Fin ∧ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁)) → {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ∈ Fin)
170	167, 168, 169	mp2an 692	. . . . . 6 ⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ∈ Fin
171	4, 170	eqeltri 2832	. . . . 5 ⊢ 𝑀 ∈ Fin
172		ssrab2 4032	. . . . 5 ⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀
173		ssfi 9097	. . . . 5 ⊢ ((𝑀 ∈ Fin ∧ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀) → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin)
174	171, 172, 173	mp2an 692	. . . 4 ⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin
175		prfi 9224	. . . . 5 ⊢ {0, 1} ∈ Fin
176		fzfid 13896	. . . . 5 ⊢ (𝜑 → (1...𝐾) ∈ Fin)
177		mapfi 9248	. . . . 5 ⊢ (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → ({0, 1} ↑_m (1...𝐾)) ∈ Fin)
178	175, 176, 177	sylancr 587	. . . 4 ⊢ (𝜑 → ({0, 1} ↑_m (1...𝐾)) ∈ Fin)
179		hashdom 14302	. . . 4 ⊢ (({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin ∧ ({0, 1} ↑_m (1...𝐾)) ∈ Fin) → ((♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1} ↑_m (1...𝐾))) ↔ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑_m (1...𝐾))))
180	174, 178, 179	sylancr 587	. . 3 ⊢ (𝜑 → ((♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1} ↑_m (1...𝐾))) ↔ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ≼ ({0, 1} ↑_m (1...𝐾))))
181	166, 180	mpbird 257	. 2 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (♯‘({0, 1} ↑_m (1...𝐾))))
182		hashmap 14358	. . . 4 ⊢ (({0, 1} ∈ Fin ∧ (1...𝐾) ∈ Fin) → (♯‘({0, 1} ↑_m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
183	175, 176, 182	sylancr 587	. . 3 ⊢ (𝜑 → (♯‘({0, 1} ↑_m (1...𝐾))) = ((♯‘{0, 1})↑(♯‘(1...𝐾))))
184		prhash2ex 14322	. . . . 5 ⊢ (♯‘{0, 1}) = 2
185	184	a1i 11	. . . 4 ⊢ (𝜑 → (♯‘{0, 1}) = 2)
186		prmrec.2	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ)
187	186	nnnn0d 12462	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
188		hashfz1 14269	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (♯‘(1...𝐾)) = 𝐾)
189	187, 188	syl 17	. . . 4 ⊢ (𝜑 → (♯‘(1...𝐾)) = 𝐾)
190	185, 189	oveq12d 7376	. . 3 ⊢ (𝜑 → ((♯‘{0, 1})↑(♯‘(1...𝐾))) = (2↑𝐾))
191	183, 190	eqtrd 2771	. 2 ⊢ (𝜑 → (♯‘({0, 1} ↑_m (1...𝐾))) = (2↑𝐾))
192	181, 191	breqtrd 5124	1 ⊢ (𝜑 → (♯‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 {crab 3399 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ifcif 4479 {cpr 4582 class class class wbr 5098 ↦ cmpt 5179 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ↑_m cmap 8763 ≼ cdom 8881 Fincfn 8883 supcsup 9343 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 < clt 11166 ≤ cle 11167 / cdiv 11794 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 ↑cexp 13984 ♯chash 14253 ∥ cdvds 16179 ℙcprime 16598 pCnt cpc 16764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-xnn0 12475 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-fz 13424 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765

This theorem is referenced by: prmreclem3 16846

W3C validator