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Theorem oddprmdvds 16241

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 16038	. . . 4 ⊢ 2 ∈ ℙ
2		pcndvds2 16206	. . . 4 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℕ) → ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
3	1, 2	mpan 688	. . 3 ⊢ (𝐾 ∈ ℕ → ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
4		pcdvds 16202	. . . 4 ⊢ ((2 ∈ ℙ ∧ 𝐾 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∥ 𝐾)
5	1, 4	mpan 688	. . 3 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∥ 𝐾)
6		2nn 11713	. . . . . . . . 9 ⊢ 2 ∈ ℕ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℕ)
8	1	a1i 11	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℙ)
9		id 22	. . . . . . . . 9 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℕ)
10	8, 9	pccld 16189	. . . . . . . 8 ⊢ (𝐾 ∈ ℕ → (2 pCnt 𝐾) ∈ ℕ₀)
11	7, 10	nnexpcld 13609	. . . . . . 7 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℕ)
12		nndivdvds 15618	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ (2↑(2 pCnt 𝐾)) ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
13	11, 12	mpdan 685	. . . . . 6 ⊢ (𝐾 ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
14	13	adantr 483	. . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((2↑(2 pCnt 𝐾)) ∥ 𝐾 ↔ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ))
15		elnn1uz2 12328	. . . . . . 7 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ ↔ ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ∨ (𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2)))
16		nncn 11648	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℂ)
17		nncn 11648	. . . . . . . . . . . . . . 15 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℂ)
18		nnne0 11674	. . . . . . . . . . . . . . 15 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → (2↑(2 pCnt 𝐾)) ≠ 0)
19	17, 18	jca 514	. . . . . . . . . . . . . 14 ⊢ ((2↑(2 pCnt 𝐾)) ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
20	11, 19	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ ℕ → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
21		3anass 1091	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
23	22	adantr 483	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (𝐾 ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
24		diveq1 11333	. . . . . . . . . . 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 17	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ ∧ ¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → ((𝐾 / (2↑(2 pCnt 𝐾))) = 1 ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
26	10	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → (2 pCnt 𝐾) ∈ ℕ₀)
27		oveq2 7166	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (2 pCnt 𝐾) → (2↑𝑛) = (2↑(2 pCnt 𝐾)))
28	27	eqeq2d 2834	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = (2 pCnt 𝐾) → (𝐾 = (2↑𝑛) ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
29	28	adantl 484	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) ∧ 𝑛 = (2 pCnt 𝐾)) → (𝐾 = (2↑𝑛) ↔ 𝐾 = (2↑(2 pCnt 𝐾))))
30		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → 𝐾 = (2↑(2 pCnt 𝐾)))
31	26, 29, 30	rspcedvd 3628	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝐾 = (2↑(2 pCnt 𝐾))) → ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛))
32	31	ex 415	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ ℕ → (𝐾 = (2↑(2 pCnt 𝐾)) → ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛)))
33		pm2.24 124	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))
34	32, 33	syl6 35	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℕ → (𝐾 = (2↑(2 pCnt 𝐾)) → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)))
35	34	adantr 483	. . . . . . . . . 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 242	. . . . . . . . 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 32	. . . . . . . 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 16050	. . . . . . . . 9 ⊢ ((𝐾 / (2↑(2 pCnt 𝐾))) ∈ (ℤ_≥‘2) → ∃𝑞 ∈ ℙ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))))
39		breq1 5071	. . . . . . . . . . . . . . . . 17 ⊢ (𝑞 = 2 → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) ↔ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
40	39	biimpcd 251	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (𝑞 = 2 → 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
41	40	adantl 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (𝑞 = 2 → 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))))
42	41	necon3bd 3032	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾)))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → 𝑞 ≠ 2))
43	42	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → (¬ 2 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) → 𝑞 ≠ 2)))
44		prmnn 16020	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ)
45	5, 13	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ ℕ → (𝐾 / (2↑(2 pCnt 𝐾))) ∈ ℕ)
46		nndivides 15619	. . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → (𝑞 ∥ (𝐾 / (2↑(2 pCnt 𝐾))) ↔ ∃𝑚 ∈ ℕ (𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾)))))
48		eqcom 2830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) ↔ (𝐾 / (2↑(2 pCnt 𝐾))) = (𝑚 · 𝑞))
49	16	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝐾 ∈ ℂ)
50		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
51	44	ad2antlr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℕ)
52	50, 51	nnmulcld 11693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (𝑚 · 𝑞) ∈ ℕ)
53	52	nncnd 11656	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (𝑚 · 𝑞) ∈ ℂ)
54	11	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℕ)
55	54, 19	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ (2↑(2 pCnt 𝐾)) ≠ 0))
56		divmul 11303	. . . . . . . . . . . . . . . . . 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 1367	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((𝐾 / (2↑(2 pCnt 𝐾))) = (𝑚 · 𝑞) ↔ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾))
58	48, 57	syl5bb 285	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((𝑚 · 𝑞) = (𝐾 / (2↑(2 pCnt 𝐾))) ↔ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾))
59		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) → 𝑞 ∈ ℙ)
60	59	adantr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℙ)
61	60	anim1i 616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → (𝑞 ∈ ℙ ∧ 𝑞 ≠ 2))
62		eldifsn 4721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑞 ∈ (ℙ ∖ {2}) ↔ (𝑞 ∈ ℙ ∧ 𝑞 ≠ 2))
63	61, 62	sylibr 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∈ (ℙ ∖ {2}))
64	63	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → 𝑞 ∈ (ℙ ∖ {2}))
65		breq1 5071	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = 𝑞 → (𝑝 ∥ 𝐾 ↔ 𝑞 ∥ 𝐾))
66	65	adantl 484	. . . . . . . . . . . . . . . . . . . 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 11693	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℕ)
68	67	nnzd 12089	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ)
69	44	nnzd 12089	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℤ)
70	69	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℤ)
71	68, 70	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ))
72	71	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → (((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ))
73		dvdsmul2 15634	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((2↑(2 pCnt 𝐾)) · 𝑚) ∈ ℤ ∧ 𝑞 ∈ ℤ) → 𝑞 ∥ (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞))
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∥ (((2↑(2 pCnt 𝐾)) · 𝑚) · 𝑞))
75		2nn0 11917	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 2 ∈ ℕ₀
76	75	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐾 ∈ ℕ → 2 ∈ ℕ₀)
77	76, 10	nn0expcld 13610	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝐾 ∈ ℕ → (2↑(2 pCnt 𝐾)) ∈ ℕ₀)
78	77	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℕ₀)
79	78	nn0cnd 11960	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → (2↑(2 pCnt 𝐾)) ∈ ℂ)
80		nncn 11648	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
81	80	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
82	44	nncnd 11656	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℂ)
83	82	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → 𝑞 ∈ ℂ)
84	79, 81, 83	3jca 1124	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ 𝑞 ∈ ℂ))
85	84	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → ((2↑(2 pCnt 𝐾)) ∈ ℂ ∧ 𝑚 ∈ ℂ ∧ 𝑞 ∈ ℂ))
86		mulass 10627	. . . . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . . . 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 5094	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) → 𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)))
89	88	adantr 483	. . . . . . . . . . . . . . . . . . . . 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 5072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾 → (𝑞 ∥ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) ↔ 𝑞 ∥ 𝐾))
91	90	adantl 484	. . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → 𝑞 ∥ 𝐾)
93	64, 66, 92	rspcedvd 3628	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐾 ∈ ℕ ∧ 𝑞 ∈ ℙ) ∧ 𝑚 ∈ ℕ) ∧ 𝑞 ≠ 2) ∧ ((2↑(2 pCnt 𝐾)) · (𝑚 · 𝑞)) = 𝐾) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾)
94	93	a1d 25	. . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . 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 3286	. . . . . . . . . . . . . 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 242	. . . . . . . . . . . . 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 72	. . . . . . . . . . . 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 3286	. . . . . . . . . . 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 32	. . . . . . . . . 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 413	. . . . . . . . 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 17	. . . . . . . 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 853	. . . . . . 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 219	. . . . . 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 32	. . . . 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 242	. . . 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 415	. . 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 49	. 2 ⊢ (𝐾 ∈ ℕ → (¬ ∃𝑛 ∈ ℕ₀ 𝐾 = (2↑𝑛) → ∃𝑝 ∈ (ℙ ∖ {2})𝑝 ∥ 𝐾))
111	110	imp 409	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 ∖ cdif 3935 {csn 4569 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 / cdiv 11299 ℕcn 11640 2c2 11695 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ↑cexp 13432 ∥ cdvds 15609 ℙcprime 16017 pCnt cpc 16175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-fz 12896 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-gcd 15846 df-prm 16018 df-pc 16176

This theorem is referenced by: 2pwp1prm 43758

W3C validator