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Theorem prmreclem1 16799

Description: Lemma for prmrec 16805. 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 4042	. . 3 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℕ
2		breq2 5114	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑟↑2) ∥ 𝑛 ↔ (𝑟↑2) ∥ 𝑁))
3	2	rabbidv 3413	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
4	3	supeq1d 9391	. . . . 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 11244	. . . . . 6 ⊢ < Or ℝ
7	6	supex 9408	. . . . 5 ⊢ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ V
8	4, 5, 7	fvmpt 6953	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
9		nnssz 12530	. . . . . 6 ⊢ ℕ ⊆ ℤ
10	1, 9	sstri 3956	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℤ
11		oveq1 7369	. . . . . . . . 9 ⊢ (𝑟 = 1 → (𝑟↑2) = (1↑2))
12		sq1 14109	. . . . . . . . 9 ⊢ (1↑2) = 1
13	11, 12	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑟↑2) = 1)
14	13	breq1d 5120	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑟↑2) ∥ 𝑁 ↔ 1 ∥ 𝑁))
15		1nn 12173	. . . . . . . 8 ⊢ 1 ∈ ℕ
16	15	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℕ)
17		nnz 12529	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
18		1dvds 16164	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∥ 𝑁)
20	14, 16, 19	elrabd 3650	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
21	20	ne0d 4300	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ≠ ∅)
22		nnz 12529	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
23		zsqcl 14044	. . . . . . . . . . 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 16203	. . . . . . . . . 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 14119	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ≤ (𝑧↑2))
29	28	adantl 482	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ≤ (𝑧↑2))
30		nnre 12169	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
31	30	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℝ)
32	31	resqcld 14040	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧↑2) ∈ ℝ)
33		nnre 12169	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
34	33	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑁 ∈ ℝ)
35		letr 11258	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ (𝑧↑2) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
36	31, 32, 34, 35	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
37	29, 36	mpand 693	. . . . . . . . 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 3139	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
40		oveq1 7369	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → (𝑟↑2) = (𝑧↑2))
41	40	breq1d 5120	. . . . . . . 8 ⊢ (𝑟 = 𝑧 → ((𝑟↑2) ∥ 𝑁 ↔ (𝑧↑2) ∥ 𝑁))
42	41	ralrab 3654	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁 ↔ ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
43	39, 42	sylibr 233	. . . . . 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 12872	. . . . 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 1466	. . . 4 ⊢ (𝑁 ∈ ℕ → sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
48	8, 47	eqeltrd 2832	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
49	1, 48	sselid 3945	. 2 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ ℕ)
50		oveq1 7369	. . . . . 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 3415	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} = {𝑧 ∈ ℕ ∣ (𝑧↑2) ∥ 𝑁}
53	51, 52	elrab2 3651	. . . 4 ⊢ ((𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ↔ ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
54	48, 53	sylib 217	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
55	54	simprd 496	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∥ 𝑁)
56	49	adantr 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℕ)
57	56	nncnd 12178	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℂ)
58	57	mulridd 11181	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 1) = (𝑄‘𝑁))
59		eluz2gt1 12854	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 1 < 𝐾)
60	59	adantl 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 < 𝐾)
61		1red 11165	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
62		eluz2nn 12818	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
63	62	adantl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ)
64	63	nnred 12177	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℝ)
65	56	nnred 12177	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℝ)
66	56	nngt0d 12211	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 0 < (𝑄‘𝑁))
67		ltmul2 12015	. . . . . . . 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 1374	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (1 < 𝐾 ↔ ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾)))
69	60, 68	mpbid 231	. . . . . 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 12186	. . . . . . . 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 12177	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 𝐾) ∈ ℝ)
74	65, 73	ltnled 11311	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) < ((𝑄‘𝑁) · 𝐾) ↔ ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁)))
75	70, 74	mpbid 231	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁))
76	45	ad2antrr 724	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
77		oveq1 7369	. . . . . . . 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 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
80		simpr 485	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
81	63	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℕ)
82	81	nnsqcld 14157	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∈ ℕ)
83		nnz 12529	. . . . . . . . . . 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 14157	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℕ)
86	9, 85	sselid 3945	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℤ)
87	85	nnne0d 12212	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ≠ 0)
88		dvdsval2 16150	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑄‘𝑁)↑2) ∥ 𝑁 ↔ (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ))
90	55, 89	mpbid 231	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
91	90	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
92	86	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℤ)
93		dvdscmul 16176	. . . . . . . . . 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 1371	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑄‘𝑁) ∈ ℂ)
97	81	nncnd 12178	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℂ)
98	96, 97	sqmuld 14073	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁) · 𝐾)↑2) = (((𝑄‘𝑁)↑2) · (𝐾↑2)))
99	98	eqcomd 2737	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) = (((𝑄‘𝑁) · 𝐾)↑2))
100		nncn 12170	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101	100	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝑁 ∈ ℂ)
102	85	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℕ)
103	102	nncnd 12178	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℂ)
104	87	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ≠ 0)
105	101, 103, 104	divcan2d 11942	. . . . . . . 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 3650	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
108		suprzub 12873	. . . . . 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 1466	. . . . 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 724	. . . . 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 814	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
113	112	ex 413	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3405 ⊆ wss 3913 ∅c0 4287 class class class wbr 5110 ↦ cmpt 5193 ‘cfv 6501 (class class class)co 7362 supcsup 9385 ℂcc 11058 ℝcr 11059 0cc0 11060 1c1 11061 · cmul 11065 < clt 11198 ≤ cle 11199 / cdiv 11821 ℕcn 12162 2c2 12217 ℤcz 12508 ℤ_≥cuz 12772 ↑cexp 13977 ∥ cdvds 16147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9387 df-inf 9388 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-div 11822 df-nn 12163 df-2 12225 df-n0 12423 df-z 12509 df-uz 12773 df-seq 13917 df-exp 13978 df-dvds 16148

This theorem is referenced by: prmreclem2 16800 prmreclem3 16801

W3C validator