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Theorem prmreclem1 16998

Description: Lemma for prmrec 17004. 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 4035	. . 3 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℕ
2		breq2 5115	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑟↑2) ∥ 𝑛 ↔ (𝑟↑2) ∥ 𝑁))
3	2	rabbidv 3425	. . . . . 6 ⊢ (𝑛 = 𝑁 → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑛} = {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
4	3	supeq1d 9413	. . . . 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 11305	. . . . . 6 ⊢ < Or ℝ
7	6	supex 9431	. . . . 5 ⊢ sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ V
8	4, 5, 7	fvmpt 6993	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) = sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ))
9		nnssz 12628	. . . . . 6 ⊢ ℕ ⊆ ℤ
10	1, 9	sstri 3947	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ⊆ ℤ
11		oveq1 7426	. . . . . . . . 9 ⊢ (𝑟 = 1 → (𝑟↑2) = (1↑2))
12		sq1 14249	. . . . . . . . 9 ⊢ (1↑2) = 1
13	11, 12	eqtrdi 2816	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑟↑2) = 1)
14	13	breq1d 5121	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑟↑2) ∥ 𝑁 ↔ 1 ∥ 𝑁))
15		1nn 12259	. . . . . . . 8 ⊢ 1 ∈ ℕ
16	15	a1i 11	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℕ)
17		nnz 12627	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
18		1dvds 16350	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 1 ∥ 𝑁)
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → 1 ∥ 𝑁)
20	14, 16, 19	elrabd 3654	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 1 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
21	20	ne0d 4295	. . . . 5 ⊢ (𝑁 ∈ ℕ → {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ≠ ∅)
22		nnz 12627	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
23		zsqcl 14183	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℤ → (𝑧↑2) ∈ ℤ)
24	22, 23	syl 18	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℕ → (𝑧↑2) ∈ ℤ)
25		id 23	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ)
26		dvdsle 16390	. . . . . . . . . 10 ⊢ (((𝑧↑2) ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → (𝑧↑2) ≤ 𝑁))
27	24, 25, 26	syl2anr 609	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → (𝑧↑2) ≤ 𝑁))
28		nnlesq 14259	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ≤ (𝑧↑2))
29	28	adantl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ≤ (𝑧↑2))
30		nnre 12255	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
31	30	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑧 ∈ ℝ)
32	31	resqcld 14179	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧↑2) ∈ ℝ)
33		nnre 12255	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
34	33	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → 𝑁 ∈ ℝ)
35		letr 11319	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ (𝑧↑2) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
36	31, 32, 34, 35	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 ≤ (𝑧↑2) ∧ (𝑧↑2) ≤ 𝑁) → 𝑧 ≤ 𝑁))
37	29, 36	mpand 708	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ≤ 𝑁 → 𝑧 ≤ 𝑁))
38	27, 37	syld 48	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
39	38	ralrimiva 3159	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
40		oveq1 7426	. . . . . . . . 9 ⊢ (𝑟 = 𝑧 → (𝑟↑2) = (𝑧↑2))
41	40	breq1d 5121	. . . . . . . 8 ⊢ (𝑟 = 𝑧 → ((𝑟↑2) ∥ 𝑁 ↔ (𝑧↑2) ∥ 𝑁))
42	41	ralrab 3659	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁 ↔ ∀𝑧 ∈ ℕ ((𝑧↑2) ∥ 𝑁 → 𝑧 ≤ 𝑁))
43	39, 42	sylibr 237	. . . . . 6 ⊢ (𝑁 ∈ ℕ → ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑁)
44		brralrspcev 5173	. . . . . 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 596	. . . . 5 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
46		suprzcl2 12978	. . . . 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 1495	. . . 4 ⊢ (𝑁 ∈ ℕ → sup({𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}, ℝ, < ) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
48	8, 47	eqeltrd 2865	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
49	1, 48	sselid 3936	. 2 ⊢ (𝑁 ∈ ℕ → (𝑄‘𝑁) ∈ ℕ)
50		oveq1 7426	. . . . . 6 ⊢ (𝑧 = (𝑄‘𝑁) → (𝑧↑2) = ((𝑄‘𝑁)↑2))
51	50	breq1d 5121	. . . . 5 ⊢ (𝑧 = (𝑄‘𝑁) → ((𝑧↑2) ∥ 𝑁 ↔ ((𝑄‘𝑁)↑2) ∥ 𝑁))
52	41	cbvrabv 3428	. . . . 5 ⊢ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} = {𝑧 ∈ ℕ ∣ (𝑧↑2) ∥ 𝑁}
53	51, 52	elrab2 3656	. . . 4 ⊢ ((𝑄‘𝑁) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁} ↔ ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
54	48, 53	sylib 221	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁) ∈ ℕ ∧ ((𝑄‘𝑁)↑2) ∥ 𝑁))
55	54	simprd 501	. 2 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∥ 𝑁)
56	49	adantr 486	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℕ)
57	56	nncnd 12264	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℂ)
58	57	mulridd 11241	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 1) = (𝑄‘𝑁))
59		eluz2gt1 12960	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 1 < 𝐾)
60	59	adantl 487	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 < 𝐾)
61		1red 11224	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 1 ∈ ℝ)
62		eluz2nn 12928	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
63	62	adantl 487	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ)
64	63	nnred 12263	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℝ)
65	56	nnred 12263	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) ∈ ℝ)
66	56	nngt0d 12300	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 0 < (𝑄‘𝑁))
67		ltmul2 12081	. . . . . . . 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 1401	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (1 < 𝐾 ↔ ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾)))
69	60, 68	mpbid 235	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 1) < ((𝑄‘𝑁) · 𝐾))
70	58, 69	eqbrtrrd 5137	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → (𝑄‘𝑁) < ((𝑄‘𝑁) · 𝐾))
71		nnmulcl 12272	. . . . . . . 8 ⊢ (((𝑄‘𝑁) ∈ ℕ ∧ 𝐾 ∈ ℕ) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
72	49, 62, 71	syl2an 608	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
73	72	nnred 12263	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) · 𝐾) ∈ ℝ)
74	65, 73	ltnled 11372	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ((𝑄‘𝑁) < ((𝑄‘𝑁) · 𝐾) ↔ ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁)))
75	70, 74	mpbid 235	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁))
76	45	ad2antrr 739	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ∃𝑥 ∈ ℤ ∀𝑧 ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁}𝑧 ≤ 𝑥)
77		oveq1 7426	. . . . . . . 8 ⊢ (𝑟 = ((𝑄‘𝑁) · 𝐾) → (𝑟↑2) = (((𝑄‘𝑁) · 𝐾)↑2))
78	77	breq1d 5121	. . . . . . 7 ⊢ (𝑟 = ((𝑄‘𝑁) · 𝐾) → ((𝑟↑2) ∥ 𝑁 ↔ (((𝑄‘𝑁) · 𝐾)↑2) ∥ 𝑁))
79	72	adantr 486	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ ℕ)
80		simpr 490	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
81	63	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℕ)
82	81	nnsqcld 14298	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∈ ℕ)
83		nnz 12627	. . . . . . . . . . 11 ⊢ ((𝐾↑2) ∈ ℕ → (𝐾↑2) ∈ ℤ)
84	82, 83	syl 18	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝐾↑2) ∈ ℤ)
85	49	nnsqcld 14298	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℕ)
86	9, 85	sselid 3936	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ∈ ℤ)
87	85	nnne0d 12301	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → ((𝑄‘𝑁)↑2) ≠ 0)
88		dvdsval2 16335	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (((𝑄‘𝑁)↑2) ∥ 𝑁 ↔ (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ))
90	55, 89	mpbid 235	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
91	90	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑁 / ((𝑄‘𝑁)↑2)) ∈ ℤ)
92	86	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℤ)
93		dvdscmul 16362	. . . . . . . . . 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 1398	. . . . . . . . 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 16	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (𝑄‘𝑁) ∈ ℂ)
97	81	nncnd 12264	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝐾 ∈ ℂ)
98	96, 97	sqmuld 14212	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁) · 𝐾)↑2) = (((𝑄‘𝑁)↑2) · (𝐾↑2)))
99	98	eqcomd 2771	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁)↑2) · (𝐾↑2)) = (((𝑄‘𝑁) · 𝐾)↑2))
100		nncn 12256	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
101	100	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → 𝑁 ∈ ℂ)
102	85	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℕ)
103	102	nncnd 12264	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ∈ ℂ)
104	87	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁)↑2) ≠ 0)
105	101, 103, 104	divcan2d 12008	. . . . . . . 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 5144	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → (((𝑄‘𝑁) · 𝐾)↑2) ∥ 𝑁)
107	78, 79, 106	elrabd 3654	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ∈ {𝑟 ∈ ℕ ∣ (𝑟↑2) ∥ 𝑁})
108		suprzub 12979	. . . . . 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 1495	. . . . 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 739	. . . . 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 5141	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) ∧ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))) → ((𝑄‘𝑁) · 𝐾) ≤ (𝑄‘𝑁))
112	75, 111	mtand 828	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (ℤ_≥‘2)) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2)))
113	112	ex 418	. 2 ⊢ (𝑁 ∈ ℕ → (𝐾 ∈ (ℤ_≥‘2) → ¬ (𝐾↑2) ∥ (𝑁 / ((𝑄‘𝑁)↑2))))
114	49, 55, 113	3jca 1146	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 {crab 3418 ⊆ wss 3906 ∅c0 4286 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 supcsup 9407 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 · cmul 11120 < clt 11258 ≤ cle 11259 / cdiv 11886 ℕcn 12248 2c2 12310 ℤcz 12606 ℤ_≥cuz 12878 ↑cexp 14115 ∥ cdvds 16332

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-inf 9410 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-n0 12520 df-z 12607 df-uz 12879 df-seq 14056 df-exp 14116 df-dvds 16333

This theorem is used by: prmreclem2 16999 prmreclem3 17000

W3C validator