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Theorem prmreclem3 15460

Description: Lemma for prmrec 15464. The main inequality established here is #𝑀 ≤ #{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} · √𝑁, where {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} is the set of squarefree numbers in 𝑀. This is demonstrated by the map 𝑦 ↦ ⟨𝑦 / (𝑄‘𝑦)↑2, (𝑄‘𝑦)⟩ where 𝑄‘𝑦 is the largest number whose square divides 𝑦. (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
prmreclem3	⊢ (𝜑 → (#‘𝑀) ≤ ((2↑𝐾) · (√‘𝑁)))

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

Proof of Theorem prmreclem3 Dummy variables 𝑥 𝑦 𝑧 𝐴 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 12633	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
2		prmrec.4	. . . . . . 7 ⊢ 𝑀 = {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛}
3		ssrab2 3650	. . . . . . 7 ⊢ {𝑛 ∈ (1...𝑁) ∣ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛} ⊆ (1...𝑁)
4	2, 3	eqsstri 3598	. . . . . 6 ⊢ 𝑀 ⊆ (1...𝑁)
5		ssfi 8065	. . . . . 6 ⊢ (((1...𝑁) ∈ Fin ∧ 𝑀 ⊆ (1...𝑁)) → 𝑀 ∈ Fin)
6	1, 4, 5	mp2an 704	. . . . 5 ⊢ 𝑀 ∈ Fin
7		hashcl 13009	. . . . 5 ⊢ (𝑀 ∈ Fin → (#‘𝑀) ∈ ℕ₀)
8	6, 7	ax-mp 5	. . . 4 ⊢ (#‘𝑀) ∈ ℕ₀
9	8	nn0rei 11180	. . 3 ⊢ (#‘𝑀) ∈ ℝ
10	9	a1i 11	. 2 ⊢ (𝜑 → (#‘𝑀) ∈ ℝ)
11		2nn 11062	. . . . . 6 ⊢ 2 ∈ ℕ
12		prmrec.2	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
13	12	nnnn0d 11228	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
14		nnexpcl 12735	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝐾 ∈ ℕ₀) → (2↑𝐾) ∈ ℕ)
15	11, 13, 14	sylancr 694	. . . . 5 ⊢ (𝜑 → (2↑𝐾) ∈ ℕ)
16	15	nnnn0d 11228	. . . 4 ⊢ (𝜑 → (2↑𝐾) ∈ ℕ₀)
17		prmrec.3	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18	17	nnrpd 11746	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ⁺)
19	18	rpsqrtcld 13998	. . . . . 6 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ⁺)
20	19	rprege0d 11755	. . . . 5 ⊢ (𝜑 → ((√‘𝑁) ∈ ℝ ∧ 0 ≤ (√‘𝑁)))
21		flge0nn0 12483	. . . . 5 ⊢ (((√‘𝑁) ∈ ℝ ∧ 0 ≤ (√‘𝑁)) → (⌊‘(√‘𝑁)) ∈ ℕ₀)
22	20, 21	syl 17	. . . 4 ⊢ (𝜑 → (⌊‘(√‘𝑁)) ∈ ℕ₀)
23	16, 22	nn0mulcld 11233	. . 3 ⊢ (𝜑 → ((2↑𝐾) · (⌊‘(√‘𝑁))) ∈ ℕ₀)
24	23	nn0red 11229	. 2 ⊢ (𝜑 → ((2↑𝐾) · (⌊‘(√‘𝑁))) ∈ ℝ)
25	15	nnred 10912	. . 3 ⊢ (𝜑 → (2↑𝐾) ∈ ℝ)
26	19	rpred 11748	. . 3 ⊢ (𝜑 → (√‘𝑁) ∈ ℝ)
27	25, 26	remulcld 9949	. 2 ⊢ (𝜑 → ((2↑𝐾) · (√‘𝑁)) ∈ ℝ)
28		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀
29		ssfi 8065	. . . . . . 7 ⊢ ((𝑀 ∈ Fin ∧ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ⊆ 𝑀) → {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin)
30	6, 28, 29	mp2an 704	. . . . . 6 ⊢ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin
31		hashcl 13009	. . . . . 6 ⊢ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin → (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ∈ ℕ₀)
32	30, 31	ax-mp 5	. . . . 5 ⊢ (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ∈ ℕ₀
33	32	nn0rei 11180	. . . 4 ⊢ (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ∈ ℝ
34	22	nn0red 11229	. . . 4 ⊢ (𝜑 → (⌊‘(√‘𝑁)) ∈ ℝ)
35		remulcl 9900	. . . 4 ⊢ (((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ∈ ℝ ∧ (⌊‘(√‘𝑁)) ∈ ℝ) → ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))) ∈ ℝ)
36	33, 34, 35	sylancr 694	. . 3 ⊢ (𝜑 → ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))) ∈ ℝ)
37		fzfi 12633	. . . . . . 7 ⊢ (1...(⌊‘(√‘𝑁))) ∈ Fin
38		xpfi 8116	. . . . . . 7 ⊢ (({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin ∧ (1...(⌊‘(√‘𝑁))) ∈ Fin) → ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))) ∈ Fin)
39	30, 37, 38	mp2an 704	. . . . . 6 ⊢ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))) ∈ Fin
40		simpr 476	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ 𝑀)
41	4, 40	sseldi 3566	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ (1...𝑁))
42		elfznn 12241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
43	41, 42	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ℕ)
44		prmreclem2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
45	44	prmreclem1 15458	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ (𝑛 ∈ (ℤ_≥‘2) → ¬ (𝑛↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))))
46	45	simp2d 1067	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦)↑2) ∥ 𝑦)
47	43, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ∥ 𝑦)
48	45	simp1d 1066	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ → (𝑄‘𝑦) ∈ ℕ)
49	43, 48	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ∈ ℕ)
50	49	nnsqcld 12891	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ∈ ℕ)
51	50	nnzd 11357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ∈ ℤ)
52	50	nnne0d 10942	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ≠ 0)
53	43	nnzd 11357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ℤ)
54		dvdsval2 14824	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑄‘𝑦)↑2) ∈ ℤ ∧ ((𝑄‘𝑦)↑2) ≠ 0 ∧ 𝑦 ∈ ℤ) → (((𝑄‘𝑦)↑2) ∥ 𝑦 ↔ (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ))
55	51, 52, 53, 54	syl3anc 1318	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (((𝑄‘𝑦)↑2) ∥ 𝑦 ↔ (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ))
56	47, 55	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ)
57		nnre 10904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
58		nngt0 10926	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ → 0 < 𝑦)
59	57, 58	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 0 < 𝑦))
60		nnre 10904	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑄‘𝑦)↑2) ∈ ℕ → ((𝑄‘𝑦)↑2) ∈ ℝ)
61		nngt0 10926	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑄‘𝑦)↑2) ∈ ℕ → 0 < ((𝑄‘𝑦)↑2))
62	60, 61	jca 553	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑄‘𝑦)↑2) ∈ ℕ → (((𝑄‘𝑦)↑2) ∈ ℝ ∧ 0 < ((𝑄‘𝑦)↑2)))
63		divgt0 10770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℝ ∧ 0 < 𝑦) ∧ (((𝑄‘𝑦)↑2) ∈ ℝ ∧ 0 < ((𝑄‘𝑦)↑2))) → 0 < (𝑦 / ((𝑄‘𝑦)↑2)))
64	59, 62, 63	syl2an 493	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∈ ℕ) → 0 < (𝑦 / ((𝑄‘𝑦)↑2)))
65	43, 50, 64	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 0 < (𝑦 / ((𝑄‘𝑦)↑2)))
66		elnnz 11264	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℕ ↔ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ ∧ 0 < (𝑦 / ((𝑄‘𝑦)↑2))))
67	56, 65, 66	sylanbrc 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℕ)
68	67	nnred 10912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℝ)
69	43	nnred 10912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ℝ)
70	17	nnred 10912	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℝ)
71	70	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑁 ∈ ℝ)
72		dvdsmul1 14841	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ ∧ ((𝑄‘𝑦)↑2) ∈ ℤ) → (𝑦 / ((𝑄‘𝑦)↑2)) ∥ ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)))
73	56, 51, 72	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∥ ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)))
74	43	nncnd 10913	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ∈ ℂ)
75	50	nncnd 10913	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ∈ ℂ)
76	74, 75, 52	divcan1d 10681	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = 𝑦)
77	73, 76	breqtrd 4609	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦)
78		dvdsle 14870	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ ∧ 𝑦 ∈ ℕ) → ((𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦 → (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑦))
79	56, 43, 78	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦 → (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑦))
80	77, 79	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑦)
81		elfzle2 12216	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ≤ 𝑁)
82	41, 81	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑦 ≤ 𝑁)
83	68, 69, 71, 80, 82	letrd 10073	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑁)
84		nnuz 11599	. . . . . . . . . . . . . 14 ⊢ ℕ = (ℤ_≥‘1)
85	67, 84	syl6eleq 2698	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ (ℤ_≥‘1))
86	17	nnzd 11357	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
87	86	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑁 ∈ ℤ)
88		elfz5 12205	. . . . . . . . . . . . 13 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ ℤ) → ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ (1...𝑁) ↔ (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑁))
89	85, 87, 88	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ (1...𝑁) ↔ (𝑦 / ((𝑄‘𝑦)↑2)) ≤ 𝑁))
90	83, 89	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ (1...𝑁))
91		breq2 4587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑦 → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑦))
92	91	notbid 307	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑦 → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑦))
93	92	ralbidv 2969	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦))
94	93, 2	elrab2 3333	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑀 ↔ (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦))
95	40, 94	sylib 207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦))
96	95	simprd 478	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦)
97	77	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦)
98		eldifi 3694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℙ)
99		prmz 15227	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℤ)
100	98, 99	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ (ℙ ∖ (1...𝐾)) → 𝑝 ∈ ℤ)
101	100	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → 𝑝 ∈ ℤ)
102	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ)
103	53	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → 𝑦 ∈ ℤ)
104		dvdstr 14856	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ∈ ℤ ∧ (𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ∧ (𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦) → 𝑝 ∥ 𝑦))
105	101, 102, 103, 104	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → ((𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ∧ (𝑦 / ((𝑄‘𝑦)↑2)) ∥ 𝑦) → 𝑝 ∥ 𝑦))
106	97, 105	mpan2d 706	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2)) → 𝑝 ∥ 𝑦))
107	106	con3d 147	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑀) ∧ 𝑝 ∈ (ℙ ∖ (1...𝐾))) → (¬ 𝑝 ∥ 𝑦 → ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
108	107	ralimdva 2945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑦 → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
109	96, 108	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2)))
110		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 / ((𝑄‘𝑦)↑2)) → (𝑝 ∥ 𝑛 ↔ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
111	110	notbid 307	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑦 / ((𝑄‘𝑦)↑2)) → (¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
112	111	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑦 / ((𝑄‘𝑦)↑2)) → (∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ 𝑛 ↔ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
113	112, 2	elrab2 3333	. . . . . . . . . . 11 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ 𝑀 ↔ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ (1...𝑁) ∧ ∀𝑝 ∈ (ℙ ∖ (1...𝐾)) ¬ 𝑝 ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
114	90, 109, 113	sylanbrc 695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ 𝑀)
115	44	prmreclem1 15458	. . . . . . . . . . . . 13 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℕ → ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ ℕ ∧ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)) ∧ (𝐴 ∈ (ℤ_≥‘2) → ¬ (𝐴↑2) ∥ ((𝑦 / ((𝑄‘𝑦)↑2)) / ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2)))))
116	115	simp2d 1067	. . . . . . . . . . . 12 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℕ → ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))
117	67, 116	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))
118	115	simp1d 1066	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ ℕ → (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ ℕ)
119	67, 118	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ ℕ)
120		elnn1uz2 11641	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ ℕ ↔ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1 ∨ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ (ℤ_≥‘2)))
121	119, 120	sylib 207	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1 ∨ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ (ℤ_≥‘2)))
122	121	ord 391	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (¬ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1 → (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ (ℤ_≥‘2)))
123	44	prmreclem1 15458	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → ((𝑄‘𝑦) ∈ ℕ ∧ ((𝑄‘𝑦)↑2) ∥ 𝑦 ∧ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ (ℤ_≥‘2) → ¬ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2)))))
124	123	simp3d 1068	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) ∈ (ℤ_≥‘2) → ¬ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
125	43, 122, 124	sylsyld 59	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (¬ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1 → ¬ ((𝑄‘(𝑦 / ((𝑄‘𝑦)↑2)))↑2) ∥ (𝑦 / ((𝑄‘𝑦)↑2))))
126	117, 125	mt4d 151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1)
127		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 / ((𝑄‘𝑦)↑2)) → (𝑄‘𝑥) = (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))))
128	127	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 / ((𝑄‘𝑦)↑2)) → ((𝑄‘𝑥) = 1 ↔ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1))
129	128	elrab 3331	. . . . . . . . . 10 ⊢ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ↔ ((𝑦 / ((𝑄‘𝑦)↑2)) ∈ 𝑀 ∧ (𝑄‘(𝑦 / ((𝑄‘𝑦)↑2))) = 1))
130	114, 126, 129	sylanbrc 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑦 / ((𝑄‘𝑦)↑2)) ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1})
131	50	nnred 10912	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ∈ ℝ)
132		dvdsle 14870	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑄‘𝑦)↑2) ∈ ℤ ∧ 𝑦 ∈ ℕ) → (((𝑄‘𝑦)↑2) ∥ 𝑦 → ((𝑄‘𝑦)↑2) ≤ 𝑦))
133	51, 43, 132	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (((𝑄‘𝑦)↑2) ∥ 𝑦 → ((𝑄‘𝑦)↑2) ≤ 𝑦))
134	47, 133	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ≤ 𝑦)
135	131, 69, 71, 134, 82	letrd 10073	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ≤ 𝑁)
136	71	recnd 9947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → 𝑁 ∈ ℂ)
137	136	sqsqrtd 14026	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((√‘𝑁)↑2) = 𝑁)
138	135, 137	breqtrrd 4611	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦)↑2) ≤ ((√‘𝑁)↑2))
139	49	nnrpd 11746	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ∈ ℝ⁺)
140	19	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (√‘𝑁) ∈ ℝ⁺)
141		rprege0 11723	. . . . . . . . . . . . . 14 ⊢ ((𝑄‘𝑦) ∈ ℝ⁺ → ((𝑄‘𝑦) ∈ ℝ ∧ 0 ≤ (𝑄‘𝑦)))
142		rprege0 11723	. . . . . . . . . . . . . 14 ⊢ ((√‘𝑁) ∈ ℝ⁺ → ((√‘𝑁) ∈ ℝ ∧ 0 ≤ (√‘𝑁)))
143		le2sq 12800	. . . . . . . . . . . . . 14 ⊢ ((((𝑄‘𝑦) ∈ ℝ ∧ 0 ≤ (𝑄‘𝑦)) ∧ ((√‘𝑁) ∈ ℝ ∧ 0 ≤ (√‘𝑁))) → ((𝑄‘𝑦) ≤ (√‘𝑁) ↔ ((𝑄‘𝑦)↑2) ≤ ((√‘𝑁)↑2)))
144	141, 142, 143	syl2an 493	. . . . . . . . . . . . 13 ⊢ (((𝑄‘𝑦) ∈ ℝ⁺ ∧ (√‘𝑁) ∈ ℝ⁺) → ((𝑄‘𝑦) ≤ (√‘𝑁) ↔ ((𝑄‘𝑦)↑2) ≤ ((√‘𝑁)↑2)))
145	139, 140, 144	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦) ≤ (√‘𝑁) ↔ ((𝑄‘𝑦)↑2) ≤ ((√‘𝑁)↑2)))
146	138, 145	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ≤ (√‘𝑁))
147	26	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (√‘𝑁) ∈ ℝ)
148	49	nnzd 11357	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ∈ ℤ)
149		flge 12468	. . . . . . . . . . . 12 ⊢ (((√‘𝑁) ∈ ℝ ∧ (𝑄‘𝑦) ∈ ℤ) → ((𝑄‘𝑦) ≤ (√‘𝑁) ↔ (𝑄‘𝑦) ≤ (⌊‘(√‘𝑁))))
150	147, 148, 149	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦) ≤ (√‘𝑁) ↔ (𝑄‘𝑦) ≤ (⌊‘(√‘𝑁))))
151	146, 150	mpbid 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ≤ (⌊‘(√‘𝑁)))
152	49, 84	syl6eleq 2698	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ∈ (ℤ_≥‘1))
153	22	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘𝑁)) ∈ ℤ)
154	153	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (⌊‘(√‘𝑁)) ∈ ℤ)
155		elfz5 12205	. . . . . . . . . . 11 ⊢ (((𝑄‘𝑦) ∈ (ℤ_≥‘1) ∧ (⌊‘(√‘𝑁)) ∈ ℤ) → ((𝑄‘𝑦) ∈ (1...(⌊‘(√‘𝑁))) ↔ (𝑄‘𝑦) ≤ (⌊‘(√‘𝑁))))
156	152, 154, 155	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ((𝑄‘𝑦) ∈ (1...(⌊‘(√‘𝑁))) ↔ (𝑄‘𝑦) ≤ (⌊‘(√‘𝑁))))
157	151, 156	mpbird 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → (𝑄‘𝑦) ∈ (1...(⌊‘(√‘𝑁))))
158		opelxpi 5072	. . . . . . . . 9 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) ∈ {𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∧ (𝑄‘𝑦) ∈ (1...(⌊‘(√‘𝑁)))) → ⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ ∈ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))))
159	130, 157, 158	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑀) → ⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ ∈ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))))
160	159	ex 449	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑀 → ⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ ∈ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))))
161		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝑦 / ((𝑄‘𝑦)↑2)) ∈ V
162		fvex 6113	. . . . . . . . . . . 12 ⊢ (𝑄‘𝑦) ∈ V
163	161, 162	opth 4871	. . . . . . . . . . 11 ⊢ (⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩ ↔ ((𝑦 / ((𝑄‘𝑦)↑2)) = (𝑧 / ((𝑄‘𝑧)↑2)) ∧ (𝑄‘𝑦) = (𝑄‘𝑧)))
164		oveq1 6556	. . . . . . . . . . . 12 ⊢ ((𝑄‘𝑦) = (𝑄‘𝑧) → ((𝑄‘𝑦)↑2) = ((𝑄‘𝑧)↑2))
165		oveq12 6558	. . . . . . . . . . . 12 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) = (𝑧 / ((𝑄‘𝑧)↑2)) ∧ ((𝑄‘𝑦)↑2) = ((𝑄‘𝑧)↑2)) → ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = ((𝑧 / ((𝑄‘𝑧)↑2)) · ((𝑄‘𝑧)↑2)))
166	164, 165	sylan2 490	. . . . . . . . . . 11 ⊢ (((𝑦 / ((𝑄‘𝑦)↑2)) = (𝑧 / ((𝑄‘𝑧)↑2)) ∧ (𝑄‘𝑦) = (𝑄‘𝑧)) → ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = ((𝑧 / ((𝑄‘𝑧)↑2)) · ((𝑄‘𝑧)↑2)))
167	163, 166	sylbi 206	. . . . . . . . . 10 ⊢ (⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩ → ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = ((𝑧 / ((𝑄‘𝑧)↑2)) · ((𝑄‘𝑧)↑2)))
168	76	adantrr 749	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → ((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = 𝑦)
169	42	ssriv 3572	. . . . . . . . . . . . . . 15 ⊢ (1...𝑁) ⊆ ℕ
170	4, 169	sstri 3577	. . . . . . . . . . . . . 14 ⊢ 𝑀 ⊆ ℕ
171		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → 𝑧 ∈ 𝑀)
172	170, 171	sseldi 3566	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → 𝑧 ∈ ℕ)
173	172	nncnd 10913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → 𝑧 ∈ ℂ)
174	44	prmreclem1 15458	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ℕ → ((𝑄‘𝑧) ∈ ℕ ∧ ((𝑄‘𝑧)↑2) ∥ 𝑧 ∧ (2 ∈ (ℤ_≥‘2) → ¬ (2↑2) ∥ (𝑧 / ((𝑄‘𝑧)↑2)))))
175	174	simp1d 1066	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ → (𝑄‘𝑧) ∈ ℕ)
176	172, 175	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → (𝑄‘𝑧) ∈ ℕ)
177	176	nnsqcld 12891	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → ((𝑄‘𝑧)↑2) ∈ ℕ)
178	177	nncnd 10913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → ((𝑄‘𝑧)↑2) ∈ ℂ)
179	177	nnne0d 10942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → ((𝑄‘𝑧)↑2) ≠ 0)
180	173, 178, 179	divcan1d 10681	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → ((𝑧 / ((𝑄‘𝑧)↑2)) · ((𝑄‘𝑧)↑2)) = 𝑧)
181	168, 180	eqeq12d 2625	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → (((𝑦 / ((𝑄‘𝑦)↑2)) · ((𝑄‘𝑦)↑2)) = ((𝑧 / ((𝑄‘𝑧)↑2)) · ((𝑄‘𝑧)↑2)) ↔ 𝑦 = 𝑧))
182	167, 181	syl5ib 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → (⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩ → 𝑦 = 𝑧))
183		id 22	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
184		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑄‘𝑦) = (𝑄‘𝑧))
185	184	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑄‘𝑦)↑2) = ((𝑄‘𝑧)↑2))
186	183, 185	oveq12d 6567	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 / ((𝑄‘𝑦)↑2)) = (𝑧 / ((𝑄‘𝑧)↑2)))
187	186, 184	opeq12d 4348	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩)
188	182, 187	impbid1 214	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀)) → (⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩ ↔ 𝑦 = 𝑧))
189	188	ex 449	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ 𝑀 ∧ 𝑧 ∈ 𝑀) → (⟨(𝑦 / ((𝑄‘𝑦)↑2)), (𝑄‘𝑦)⟩ = ⟨(𝑧 / ((𝑄‘𝑧)↑2)), (𝑄‘𝑧)⟩ ↔ 𝑦 = 𝑧)))
190	160, 189	dom2d 7882	. . . . . 6 ⊢ (𝜑 → (({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))) ∈ Fin → 𝑀 ≼ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))))
191	39, 190	mpi 20	. . . . 5 ⊢ (𝜑 → 𝑀 ≼ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))))
192		hashdom 13029	. . . . . 6 ⊢ ((𝑀 ∈ Fin ∧ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))) ∈ Fin) → ((#‘𝑀) ≤ (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))) ↔ 𝑀 ≼ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))))
193	6, 39, 192	mp2an 704	. . . . 5 ⊢ ((#‘𝑀) ≤ (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))) ↔ 𝑀 ≼ ({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁)))))
194	191, 193	sylibr 223	. . . 4 ⊢ (𝜑 → (#‘𝑀) ≤ (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))))
195		hashxp 13081	. . . . . 6 ⊢ (({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} ∈ Fin ∧ (1...(⌊‘(√‘𝑁))) ∈ Fin) → (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))) = ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (#‘(1...(⌊‘(√‘𝑁))))))
196	30, 37, 195	mp2an 704	. . . . 5 ⊢ (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))) = ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (#‘(1...(⌊‘(√‘𝑁)))))
197		hashfz1 12996	. . . . . . 7 ⊢ ((⌊‘(√‘𝑁)) ∈ ℕ₀ → (#‘(1...(⌊‘(√‘𝑁)))) = (⌊‘(√‘𝑁)))
198	22, 197	syl 17	. . . . . 6 ⊢ (𝜑 → (#‘(1...(⌊‘(√‘𝑁)))) = (⌊‘(√‘𝑁)))
199	198	oveq2d 6565	. . . . 5 ⊢ (𝜑 → ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (#‘(1...(⌊‘(√‘𝑁))))) = ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))))
200	196, 199	syl5eq 2656	. . . 4 ⊢ (𝜑 → (#‘({𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1} × (1...(⌊‘(√‘𝑁))))) = ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))))
201	194, 200	breqtrd 4609	. . 3 ⊢ (𝜑 → (#‘𝑀) ≤ ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))))
202	33	a1i 11	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ∈ ℝ)
203	22	nn0ge0d 11231	. . . 4 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘𝑁)))
204		prmrec.1	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (1 / 𝑛), 0))
205	204, 12, 17, 2, 44	prmreclem2 15459	. . . 4 ⊢ (𝜑 → (#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) ≤ (2↑𝐾))
206	202, 25, 34, 203, 205	lemul1ad 10842	. . 3 ⊢ (𝜑 → ((#‘{𝑥 ∈ 𝑀 ∣ (𝑄‘𝑥) = 1}) · (⌊‘(√‘𝑁))) ≤ ((2↑𝐾) · (⌊‘(√‘𝑁))))
207	10, 36, 24, 201, 206	letrd 10073	. 2 ⊢ (𝜑 → (#‘𝑀) ≤ ((2↑𝐾) · (⌊‘(√‘𝑁))))
208	15	nnrpd 11746	. . . 4 ⊢ (𝜑 → (2↑𝐾) ∈ ℝ⁺)
209	208	rprege0d 11755	. . 3 ⊢ (𝜑 → ((2↑𝐾) ∈ ℝ ∧ 0 ≤ (2↑𝐾)))
210		fllelt 12460	. . . . 5 ⊢ ((√‘𝑁) ∈ ℝ → ((⌊‘(√‘𝑁)) ≤ (√‘𝑁) ∧ (√‘𝑁) < ((⌊‘(√‘𝑁)) + 1)))
211	26, 210	syl 17	. . . 4 ⊢ (𝜑 → ((⌊‘(√‘𝑁)) ≤ (√‘𝑁) ∧ (√‘𝑁) < ((⌊‘(√‘𝑁)) + 1)))
212	211	simpld 474	. . 3 ⊢ (𝜑 → (⌊‘(√‘𝑁)) ≤ (√‘𝑁))
213		lemul2a 10757	. . 3 ⊢ ((((⌊‘(√‘𝑁)) ∈ ℝ ∧ (√‘𝑁) ∈ ℝ ∧ ((2↑𝐾) ∈ ℝ ∧ 0 ≤ (2↑𝐾))) ∧ (⌊‘(√‘𝑁)) ≤ (√‘𝑁)) → ((2↑𝐾) · (⌊‘(√‘𝑁))) ≤ ((2↑𝐾) · (√‘𝑁)))
214	34, 26, 209, 212, 213	syl31anc 1321	. 2 ⊢ (𝜑 → ((2↑𝐾) · (⌊‘(√‘𝑁))) ≤ ((2↑𝐾) · (√‘𝑁)))
215	10, 24, 27, 207, 214	letrd 10073	1 ⊢ (𝜑 → (#‘𝑀) ≤ ((2↑𝐾) · (√‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ifcif 4036 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ≼ cdom 7839 Fincfn 7841 supcsup 8229 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708 ...cfz 12197 ⌊cfl 12453 ↑cexp 12722 #chash 12979 √csqrt 13821 ∥ cdvds 14821 ℙcprime 15223

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380

This theorem is referenced by: prmreclem5 15462

W3C validator