fmtnoprmfac1 - Mathbox for Alexander van der Vekens

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Theorem fmtnoprmfac1 46233

Description: Divisor of Fermat number (special form of Euler's result, see fmtnofac1 46238): Let F_n be a Fermat number. Let p be a prime divisor of F_n. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021.)

**Assertion**
Ref	Expression
fmtnoprmfac1	⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Distinct variable groups: 𝑘,𝑁 𝑃,𝑘

**Proof of Theorem fmtnoprmfac1**
Step	Hyp	Ref	Expression
1		breq1 5152	. . . . . . 7 ⊢ (𝑃 = 2 → (𝑃 ∥ (FermatNo‘𝑁) ↔ 2 ∥ (FermatNo‘𝑁)))
2	1	adantr 482	. . . . . 6 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (FermatNo‘𝑁) ↔ 2 ∥ (FermatNo‘𝑁)))
3		nnnn0 12479	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
4		fmtnoodd 46201	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → ¬ 2 ∥ (FermatNo‘𝑁))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ (FermatNo‘𝑁))
6	5	adantl 483	. . . . . . 7 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → ¬ 2 ∥ (FermatNo‘𝑁))
7	6	pm2.21d 121	. . . . . 6 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → (2 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
8	2, 7	sylbid 239	. . . . 5 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
9	8	a1d 25	. . . 4 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → (𝑃 ∈ ℙ → (𝑃 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
10	9	ex 414	. . 3 ⊢ (𝑃 = 2 → (𝑁 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))))
11	10	3impd 1349	. 2 ⊢ (𝑃 = 2 → ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
12		simpr1 1195	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑁 ∈ ℕ)
13		neqne 2949	. . . . . . . . . 10 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2)
14	13	anim2i 618	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
15		eldifsn 4791	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
16	14, 15	sylibr 233	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → 𝑃 ∈ (ℙ ∖ {2}))
17	16	ex 414	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ (ℙ ∖ {2})))
18	17	3ad2ant2 1135	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → (¬ 𝑃 = 2 → 𝑃 ∈ (ℙ ∖ {2})))
19	18	impcom 409	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑃 ∈ (ℙ ∖ {2}))
20		simpr3 1197	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑃 ∥ (FermatNo‘𝑁))
21		fmtnoprmfac1lem 46232	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
22	12, 19, 20, 21	syl3anc 1372	. . . 4 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
23		prmnn 16611	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
24	23	ad2antll 728	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 𝑃 ∈ ℕ)
25		2z 12594	. . . . . . . 8 ⊢ 2 ∈ ℤ
26	25	a1i 11	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ∈ ℤ)
27	13	necomd 2997	. . . . . . . . 9 ⊢ (¬ 𝑃 = 2 → 2 ≠ 𝑃)
28	27	adantr 482	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ≠ 𝑃)
29		2prm 16629	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℙ)
31	30	anim1i 616	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 ∈ ℙ ∧ 𝑃 ∈ ℙ))
32	31	adantl 483	. . . . . . . . 9 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (2 ∈ ℙ ∧ 𝑃 ∈ ℙ))
33		prmrp 16649	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((2 gcd 𝑃) = 1 ↔ 2 ≠ 𝑃))
34	32, 33	syl 17	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((2 gcd 𝑃) = 1 ↔ 2 ≠ 𝑃))
35	28, 34	mpbird 257	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (2 gcd 𝑃) = 1)
36		odzphi 16729	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
37	24, 26, 35, 36	syl3anc 1372	. . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
38		phiprm 16710	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
39	38	ad2antll 728	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (ϕ‘𝑃) = (𝑃 − 1))
40	39	breq2d 5161	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃) ↔ ((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1)))
41		breq1 5152	. . . . . . . . . . 11 ⊢ (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) ↔ (2↑(𝑁 + 1)) ∥ (𝑃 − 1)))
42	41	adantl 483	. . . . . . . . . 10 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) ↔ (2↑(𝑁 + 1)) ∥ (𝑃 − 1)))
43		2nn 12285	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
44	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
45		peano2nn 12224	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
46	45	nnnn0d 12532	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
47	44, 46	nnexpcld 14208	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
48	23	nnnn0d 12532	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
49		prmuz2 16633	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
50		eluzle 12835	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
51	49, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 2 ≤ 𝑃)
52		nn0ge2m1nn 12541	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℕ₀ ∧ 2 ≤ 𝑃) → (𝑃 − 1) ∈ ℕ)
53	48, 51, 52	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ)
54	47, 53	anim12i 614	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2↑(𝑁 + 1)) ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ))
55	54	adantl 483	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((2↑(𝑁 + 1)) ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ))
56		nndivides 16207	. . . . . . . . . . . . 13 ⊢ (((2↑(𝑁 + 1)) ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ) → ((2↑(𝑁 + 1)) ∥ (𝑃 − 1) ↔ ∃𝑘 ∈ ℕ (𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1)))
57	55, 56	syl 17	. . . . . . . . . . . 12 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((2↑(𝑁 + 1)) ∥ (𝑃 − 1) ↔ ∃𝑘 ∈ ℕ (𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1)))
58		eqcom 2740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) ↔ (𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))))
59	58	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) ↔ (𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1)))))
60	23	nncnd 12228	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
61	60	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℂ)
62	61	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 𝑃 ∈ ℂ)
63		1cnd 11209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
64		nncn 12220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
65	64	adantl 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
66		peano2nn0 12512	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
67	3, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
68	44, 67	nnexpcld 14208	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
69	68	nncnd 12228	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℂ)
70	69	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2↑(𝑁 + 1)) ∈ ℂ)
71	70	adantr 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → (2↑(𝑁 + 1)) ∈ ℂ)
72	65, 71	mulcld 11234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → (𝑘 · (2↑(𝑁 + 1))) ∈ ℂ)
73	62, 63, 72	subadd2d 11590	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → ((𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))) ↔ ((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃))
74	73	adantll 713	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → ((𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))) ↔ ((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃))
75		eqcom 2740	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃 ↔ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
76	75	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → (((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃 ↔ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
77	59, 74, 76	3bitrd 305	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) ↔ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
78	77	rexbidva 3177	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (∃𝑘 ∈ ℕ (𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) ↔ ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
79	78	biimpd 228	. . . . . . . . . . . 12 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (∃𝑘 ∈ ℕ (𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
80	57, 79	sylbid 239	. . . . . . . . . . 11 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((2↑(𝑁 + 1)) ∥ (𝑃 − 1) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
81	80	adantr 482	. . . . . . . . . 10 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) → ((2↑(𝑁 + 1)) ∥ (𝑃 − 1) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
82	42, 81	sylbid 239	. . . . . . . . 9 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1))) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
83	82	ex 414	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
84	83	com23 86	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) → (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
85	40, 84	sylbid 239	. . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃) → (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))))
86	37, 85	mpd 15	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
87	86	3adantr3 1172	. . . 4 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
88	22, 87	mpd 15	. . 3 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
89	88	ex 414	. 2 ⊢ (¬ 𝑃 = 2 → ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
90	11, 89	pm2.61i 182	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∖ cdif 3946 {csn 4629 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 1c1 11111 + caddc 11113 · cmul 11115 ≤ cle 11249 − cmin 11444 ℕcn 12212 2c2 12267 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ↑cexp 14027 ∥ cdvds 16197 gcd cgcd 16435 ℙcprime 16608 od_ℤcodz 16696 ϕcphi 16697 FermatNocfmtno 46195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436 df-prm 16609 df-odz 16698 df-phi 16699 df-pc 16770 df-fmtno 46196

This theorem is referenced by: fmtnoprmfac2lem1 46234

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