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Theorem oddprmdvds 16963

Description: Every positive integer which is not a power of two is divisible by an odd prime number. (Contributed by AV, 6-Aug-2021.)

**Assertion**
Ref	Expression
oddprmdvds	⊢ ((𝐾 ∈ ℕ ∧ ¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛)) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)

Distinct variable group: 𝑛,𝐾,𝑝

Proof of Theorem oddprmdvds Dummy variables 𝑚 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2prm 16750	. . . 4 ⊢ 2 ∈ ℙ
2		pcndvds2 16928	. . . 4 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
3	1, 2	mpan 702	. . 3 ⊢ (𝐾 ∈ ℕ → ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
4		pcdvds 16924	. . . 4 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∥ 𝐾)
5	1, 4	mpan 702	. . 3 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∥ 𝐾)
6		2nn 12314	. . . . . . . . 9 ⊢ 2 ∈ ℕ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℕ)
8	1	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℙ)
9		id 23	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
10	8, 9	pccld 16910	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (2 pCnt 𝐾) ∈ ℕ₀)
11	7, 10	nnexpcld 14281	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℕ)
12		nndivdvds 16319	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (2↑(2 pCnt 𝐾)) ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
13	11, 12	mpdan 699	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
14	13	adantr 485	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
15		elnn1uz2 12949	. . . . . . 7 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ ↔ ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ∨ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2)))
16		nncn 12241	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
17		nncn 12241	. . . . . . . . . . . . . . 15 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℂ)
18		nnne0 12270	. . . . . . . . . . . . . . 15 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → (2↑(2 pCnt 𝐾)) ≠ 0)
19	17, 18	jca 520	. . . . . . . . . . . . . 14 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
20	11, 19	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
21		3anass 1109	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0) ↔ (𝐾 ∈ ℂ ∧ ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0)))
22	16, 20, 21	sylanbrc 594	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
23	22	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
24		diveq1 11901	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0) → ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
25	23, 24	syl 18	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
26	10	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → (2 pCnt 𝐾) ∈ ℕ₀)
27		oveq2 7419	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 pCnt 𝐾) → (2↑𝑛) = (2↑(2 pCnt 𝐾)))
28	27	eqeq2d 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 pCnt 𝐾) → (𝐾 = (2↑𝑛) ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
29	28	adantl 486	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) ∧ 𝑛 = (2 pCnt 𝐾)) → (𝐾 = (2↑𝑛) ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
30		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → 𝐾 = (2↑(2 pCnt 𝐾)))
31	26, 29, 30	rspcedvd 3592	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛))
32	31	ex 417	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (𝐾 = (2↑(2 pCnt 𝐾)) → ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛)))
33		pm2.24 125	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))
34	32, 33	syl6 36	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → (𝐾 = (2↑(2 pCnt 𝐾)) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
35	34	adantr 485	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (𝐾 = (2↑(2 pCnt 𝐾)) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
36	25, 35	sylbid 243	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
37	36	com12 33	. . . . . . . 8 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 → ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
38		exprmfct 16763	. . . . . . . . 9 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2) → ∃𝑞 ∈ ℙ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
39		breq1 5116	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 2 → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) ↔ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
40	39	biimpcd 252	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (𝑞 = 2 → 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
41	40	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (𝑞 = 2 → 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
42	41	necon3bd 2978	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → 𝑞 ≠ 2))
43	42	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → 𝑞 ≠ 2)))
44		prmnn 16732	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
45	5, 13	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ → (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ)
46		nndivides 16320	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 ∈ ℕ ∧ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) ↔ ∃𝑚 ∈ ℕ (𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾)))))
47	44, 45, 46	syl2anr 608	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) ↔ ∃𝑚 ∈ ℕ (𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾)))))
48		eqcom 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) ↔ (𝐾 / (2↑(2 pCnt 𝐾))) = (𝑚 · 𝑞))
49	16	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝐾 ∈ ℂ)
50		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
51	44	ad2antlr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℕ)
52	50, 51	nnmulcld 12289	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (𝑚 · 𝑞) ∈ ℕ)
53	52	nncnd 12249	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (𝑚 · 𝑞) ∈ ℂ)
54	11	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℕ)
55	54, 19	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
56		divmul 11875	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ ℂ ∧ (𝑚 · 𝑞) ∈ ℂ ∧ ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0)) → ((𝐾 / (2↑(2 pCnt 𝐾))) = (𝑚 · 𝑞) ↔ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾))
57	49, 53, 55, 56	syl3anc 1396	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((𝐾 / (2↑(2 pCnt 𝐾))) = (𝑚 · 𝑞) ↔ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾))
58	48, 57	bitrid 286	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) ↔ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾))
59		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
60	59	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℙ)
61	60	anim1i 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → (𝑞 ∈ ℙ ∧ 𝑞 ≠ 2))
62		eldifsn 4758	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑞 ∈ (ℙ ∖ {2}) ↔ (𝑞 ∈ ℙ ∧ 𝑞 ≠ 2))
63	61, 62	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∈ (ℙ ∖ {2}))
64	63	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → 𝑞 ∈ (ℙ ∖ {2}))
65		breq1 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝐾 ↔ 𝑞 ∥ 𝐾))
66	65	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) ∧ 𝑝 = 𝑞) → (𝑝 ∥ 𝐾 ↔ 𝑞 ∥ 𝐾))
67	54, 50	nnmulcld 12289	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℕ)
68	67	nnzd 12617	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ)
69	44	nnzd 12617	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
70	69	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℤ)
71	68, 70	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ))
72	71	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → (((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ))
73		dvdsmul2 16336	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ) → 𝑞 ∥ (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞))
74	72, 73	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∥ (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞))
75		2nn0 12521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 2 ∈ ℕ₀
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℕ₀)
77	76, 10	nn0expcld 14282	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℕ₀)
78	77	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℕ₀)
79	78	nn0cnd 12567	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℂ)
80		nncn 12241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
81	80	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
82	44	nncnd 12249	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℂ)
83	82	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℂ)
84	79, 81, 83	3jca 1144	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ 𝑞 ∈ ℂ))
85	84	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ 𝑞 ∈ ℂ))
86		mulass 11188	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ 𝑞 ∈ ℂ) → (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞) = ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)))
87	85, 86	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞) = ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)))
88	74, 87	breqtrd 5141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)))
89	88	adantr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → 𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)))
90		breq2 5117	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾 → (𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) ↔ 𝑞 ∥ 𝐾))
91	90	adantl 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → (𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) ↔ 𝑞 ∥ 𝐾))
92	89, 91	mpbid 235	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → 𝑞 ∥ 𝐾)
93	64, 66, 92	rspcedvd 3592	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)
94	93	a1d 26	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))
95	94	exp31 424	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (𝑞 ≠ 2 → (((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
96	95	com23 87	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾 → (𝑞 ≠ 2 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
97	58, 96	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) → (𝑞 ≠ 2 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
98	97	rexlimdva 3172	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (∃𝑚 ∈ ℕ (𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) → (𝑞 ≠ 2 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
99	47, 98	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (𝑞 ≠ 2 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
100	43, 99	syldd 73	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
101	100	rexlimdva 3172	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → (∃𝑞 ∈ ℙ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
102	101	com12 33	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ℙ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (𝐾 ∈ ℕ → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
103	102	impd 415	. . . . . . . . 9 ⊢ (∃𝑞 ∈ ℙ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
104	38, 103	syl 18	. . . . . . . 8 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2) → ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
105	37, 104	jaoi 870	. . . . . . 7 ⊢ (((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ∨ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2)) → ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
106	15, 105	sylbi 220	. . . . . 6 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ → ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
107	106	com12 33	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
108	14, 107	sylbid 243	. . . 4 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
109	108	ex 417	. . 3 ⊢ (𝐾 ∈ ℕ → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))))
110	3, 5, 109	mp2d 50	. 2 ⊢ (𝐾 ∈ ℕ → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))
111	110	imp 411	1 ⊢ ((𝐾 ∈ ℕ ∧ ¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛)) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∃wrex 3095 ∖ cdif 3910 {csn 4594 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 0cc0 11100 1c1 11101 · cmul 11105 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ↑cexp 14097 ∥ cdvds 16310 ℙcprime 16729 pCnt cpc 16896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-inf 9403 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-fz 13536 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553 df-prm 16730 df-pc 16897

This theorem is referenced by: 2pwp1prm 48230

W3C validator