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Theorem prmreclem1 16894

Description: Lemma for prmrec 16900. Properties of the "square part" function, which extracts the 𝑚 of the decomposition 𝑁 = 𝑟𝑚↑2, with 𝑚 maximal and 𝑟 squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)

**Hypothesis**
Ref	Expression
prmreclem1.1	⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))

**Assertion**
Ref	Expression
prmreclem1	⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁 ∧ (𝐾 ∈ (ℤ_≥‘2) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))))

Distinct variable groups: 𝐾,𝑟 𝑛,𝑟,𝑁 𝑄,𝑟 Allowed substitution hints: 𝑄(𝑛) 𝐾(𝑛)

Proof of Theorem prmreclem1 Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4046	. . 3 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℕ
2		breq2 5114	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑟↑2) ∥ 𝑛 ↔ (𝑟↑2) ∥ 𝑁))
3	2	rabbidv 3416	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
4	3	supeq1d 9404	. . . . 5 ⊢ (𝑛 = 𝑁 → sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
5		prmreclem1.1	. . . . 5 ⊢ 𝑄 = (𝑛 ∈ ℕ ↦ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛}, ℝ, < ))
6		ltso 11261	. . . . . 6 ⊢ < Or ℝ
7	6	supex 9422	. . . . 5 ⊢ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ V
8	4, 5, 7	fvmpt 6971	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
9		nnssz 12558	. . . . . 6 ⊢ ℕ ⊆ ℤ
10	1, 9	sstri 3959	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℤ
11		oveq1 7397	. . . . . . . . 9 ⊢ (𝑟 = 1 → (𝑟↑2) = (1↑2))
12		sq1 14167	. . . . . . . . 9 ⊢ (1↑2) = 1
13	11, 12	eqtrdi 2781	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑟↑2) = 1)
14	13	breq1d 5120	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑟↑2) ∥ 𝑁 ↔ 1 ∥ 𝑁))
15		1nn 12204	. . . . . . . 8 ⊢ 1 ∈ ℕ
16	15	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℕ)
17		nnz 12557	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
18		1dvds 16247	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∥ 𝑁)
20	14, 16, 19	elrabd 3664	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
21	20	ne0d 4308	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ≠ ∅)
22		nnz 12557	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
23		zsqcl 14101	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℤ)
24	22, 23	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → (𝑧↑2) ∈ ℤ)
25		id 22	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
26		dvdsle 16287	. . . . . . . . . 10 ⊢ (((𝑧↑2) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → (𝑧↑2) ≤ 𝑁))
27	24, 25, 26	syl2anr 597	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → (𝑧↑2) ≤ 𝑁))
28		nnlesq 14177	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ≤ (𝑧↑2))
29	28	adantl 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ≤ (𝑧↑2))
30		nnre 12200	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
31	30	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℝ)
32	31	resqcld 14097	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧↑2) ∈ ℝ)
33		nnre 12200	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
34	33	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑁 ∈ ℝ)
35		letr 11275	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ (𝑧↑2) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
36	31, 32, 34, 35	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
37	29, 36	mpand 695	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ≤ 𝑁 → 𝑧 ≤ 𝑁))
38	27, 37	syld 47	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
39	38	ralrimiva 3126	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
40		oveq1 7397	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → (𝑟↑2) = (𝑧↑2))
41	40	breq1d 5120	. . . . . . . 8 ⊢ (𝑟 = 𝑧 → ((𝑟↑2) ∥ 𝑁 ↔ (𝑧↑2) ∥ 𝑁))
42	41	ralrab 3668	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁 ↔ ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
43	39, 42	sylibr 234	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁)
44		brralrspcev 5170	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁) → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
45	17, 43, 44	syl2anc 584	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
46		suprzcl2 12904	. . . . 5 ⊢ (({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℤ ∧ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥) → sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
47	10, 21, 45, 46	mp3an2i 1468	. . . 4 ⊢ (𝑁 ∈ ℕ → sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
48	8, 47	eqeltrd 2829	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
49	1, 48	sselid 3947	. 2 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ ℕ)
50		oveq1 7397	. . . . . 6 ⊢ (𝑧 = (𝑄‘𝑁) → (𝑧↑2) = ((𝑄‘𝑁)↑2))
51	50	breq1d 5120	. . . . 5 ⊢ (𝑧 = (𝑄‘𝑁) → ((𝑧↑2) ∥ 𝑁 ↔ ((𝑄‘𝑁)↑2) ∥ 𝑁))
52	41	cbvrabv 3419	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} = {𝑧 ∈ ℕ ∣ (𝑧↑2) ∥ 𝑁}
53	51, 52	elrab2 3665	. . . 4 ⊢ ((𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ↔ ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
54	48, 53	sylib 218	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
55	54	simprd 495	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∥ 𝑁)
56	49	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℕ)
57	56	nncnd 12209	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℂ)
58	57	mulridd 11198	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 1) = (𝑄‘𝑁))
59		eluz2gt1 12886	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 1 < 𝐾)
60	59	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 < 𝐾)
61		1red 11182	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
62		eluz2nn 12854	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
63	62	adantl 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ)
64	63	nnred 12208	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℝ)
65	56	nnred 12208	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℝ)
66	56	nngt0d 12242	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 0 < (𝑄‘𝑁))
67		ltmul2 12040	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ ((𝑄‘𝑁) ∈ ℝ ∧ 0 < (𝑄‘𝑁))) → (1 < 𝐾 ↔ ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾)))
68	61, 64, 65, 66, 67	syl112anc 1376	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (1 < 𝐾 ↔ ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾)))
69	60, 68	mpbid 232	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾))
70	58, 69	eqbrtrrd 5134	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) < ((𝑄‘𝑁) · 𝐾))
71		nnmulcl 12217	. . . . . . . 8 ⊢ (((𝑄‘𝑁) ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
72	49, 62, 71	syl2an 596	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
73	72	nnred 12208	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 𝐾) ∈ ℝ)
74	65, 73	ltnled 11328	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) < ((𝑄‘𝑁) · 𝐾) ↔ ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁)))
75	70, 74	mpbid 232	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁))
76	45	ad2antrr 726	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
77		oveq1 7397	. . . . . . . 8 ⊢ (𝑟 = ((𝑄‘𝑁) · 𝐾) → (𝑟↑2) = (((𝑄‘𝑁) · 𝐾)↑2))
78	77	breq1d 5120	. . . . . . 7 ⊢ (𝑟 = ((𝑄‘𝑁) · 𝐾) → ((𝑟↑2) ∥ 𝑁 ↔ (((𝑄‘𝑁) · 𝐾)↑2) ∥ 𝑁))
79	72	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
80		simpr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
81	63	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℕ)
82	81	nnsqcld 14216	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∈ ℕ)
83		nnz 12557	. . . . . . . . . . 11 ⊢ ((𝐾↑2) ∈ ℕ → (𝐾↑2) ∈ ℤ)
84	82, 83	syl 17	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∈ ℤ)
85	49	nnsqcld 14216	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℕ)
86	9, 85	sselid 3947	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℤ)
87	85	nnne0d 12243	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ≠ 0)
88		dvdsval2 16232	. . . . . . . . . . . . 13 ⊢ ((((𝑄‘𝑁)↑2) ∈ ℤ ∧ ((𝑄‘𝑁)↑2) ≠ 0 ∧ 𝑁 ∈ ℤ) → (((𝑄‘𝑁)↑2) ∥ 𝑁 ↔ (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ))
89	86, 87, 17, 88	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑄‘𝑁)↑2) ∥ 𝑁 ↔ (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ))
90	55, 89	mpbid 232	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
91	90	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
92	86	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℤ)
93		dvdscmul 16259	. . . . . . . . . 10 ⊢ (((𝐾↑2) ∈ ℤ ∧ (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ ∧ ((𝑄‘𝑁)↑2) ∈ ℤ) → ((𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) ∥ (((𝑄‘𝑁)↑2) · (𝑁 / ((𝑄‘𝑁)↑2)))))
94	84, 91, 92, 93	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) ∥ (((𝑄‘𝑁)↑2) · (𝑁 / ((𝑄‘𝑁)↑2)))))
95	80, 94	mpd 15	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) ∥ (((𝑄‘𝑁)↑2) · (𝑁 / ((𝑄‘𝑁)↑2))))
96	57	adantr 480	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑄‘𝑁) ∈ ℂ)
97	81	nncnd 12209	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℂ)
98	96, 97	sqmuld 14130	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁) · 𝐾)↑2) = (((𝑄‘𝑁)↑2) · (𝐾↑2)))
99	98	eqcomd 2736	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) = (((𝑄‘𝑁) · 𝐾)↑2))
100		nncn 12201	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101	100	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝑁 ∈ ℂ)
102	85	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℕ)
103	102	nncnd 12209	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℂ)
104	87	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ≠ 0)
105	101, 103, 104	divcan2d 11967	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁)↑2) · (𝑁 / ((𝑄‘𝑁)↑2))) = 𝑁)
106	95, 99, 105	3brtr3d 5141	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁) · 𝐾)↑2) ∥ 𝑁)
107	78, 79, 106	elrabd 3664	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
108		suprzub 12905	. . . . . 6 ⊢ (({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℤ ∧ ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥 ∧ ((𝑄‘𝑁) · 𝐾) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}) → ((𝑄‘𝑁) · 𝐾) ≤ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
109	10, 76, 107, 108	mp3an2i 1468	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ≤ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
110	8	ad2antrr 726	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑄‘𝑁) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
111	109, 110	breqtrrd 5138	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁))
112	75, 111	mtand 815	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
113	112	ex 412	. 2 ⊢ (𝑁 ∈ ℕ → (𝐾 ∈ (ℤ_≥‘2) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))))
114	49, 55, 113	3jca 1128	1 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁 ∧ (𝐾 ∈ (ℤ_≥‘2) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 ⊆ wss 3917 ∅c0 4299 class class class wbr 5110 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 supcsup 9398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 < clt 11215 ≤ cle 11216 / cdiv 11842 ℕcn 12193 2c2 12248 ℤcz 12536 ℤ_≥cuz 12800 ↑cexp 14033 ∥ cdvds 16229

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-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 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

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-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-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-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 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-n0 12450 df-z 12537 df-uz 12801 df-seq 13974 df-exp 14034 df-dvds 16230

This theorem is referenced by: prmreclem2 16895 prmreclem3 16896

W3C validator