fmtnoprmfac1 - Mathbox for Alexander van der Vekens

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Theorem fmtnoprmfac1 45435

Description: Divisor of Fermat number (special form of Euler's result, see fmtnofac1 45440): 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 5096	. . . . . . 7 ⊢ (𝑃 = 2 → (𝑃 ∥ (FermatNo‘𝑁) ↔ 2 ∥ (FermatNo‘𝑁)))
2	1	adantr 481	. . . . . 6 ⊢ ((𝑃 = 2 ∧ 𝑁 ∈ ℕ) → (𝑃 ∥ (FermatNo‘𝑁) ↔ 2 ∥ (FermatNo‘𝑁)))
3		nnnn0 12342	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
4		fmtnoodd 45403	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → ¬ 2 ∥ (FermatNo‘𝑁))
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ¬ 2 ∥ (FermatNo‘𝑁))
6	5	adantl 482	. . . . . . 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 413	. . 3 ⊢ (𝑃 = 2 → (𝑁 ∈ ℕ → (𝑃 ∈ ℙ → (𝑃 ∥ (FermatNo‘𝑁) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))))
11	10	3impd 1347	. 2 ⊢ (𝑃 = 2 → ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
12		simpr1 1193	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑁 ∈ ℕ)
13		neqne 2948	. . . . . . . . . 10 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2)
14	13	anim2i 617	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
15		eldifsn 4735	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
16	14, 15	sylibr 233	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → 𝑃 ∈ (ℙ ∖ {2}))
17	16	ex 413	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (¬ 𝑃 = 2 → 𝑃 ∈ (ℙ ∖ {2})))
18	17	3ad2ant2 1133	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → (¬ 𝑃 = 2 → 𝑃 ∈ (ℙ ∖ {2})))
19	18	impcom 408	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑃 ∈ (ℙ ∖ {2}))
20		simpr3 1195	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑃 ∥ (FermatNo‘𝑁))
21		fmtnoprmfac1lem 45434	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
22	12, 19, 20, 21	syl3anc 1370	. . . 4 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
23		prmnn 16477	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
24	23	ad2antll 726	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 𝑃 ∈ ℕ)
25		2z 12454	. . . . . . . 8 ⊢ 2 ∈ ℤ
26	25	a1i 11	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ∈ ℤ)
27	13	necomd 2996	. . . . . . . . 9 ⊢ (¬ 𝑃 = 2 → 2 ≠ 𝑃)
28	27	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ≠ 𝑃)
29		2prm 16495	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
30	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℙ)
31	30	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2 ∈ ℙ ∧ 𝑃 ∈ ℙ))
32	31	adantl 482	. . . . . . . . 9 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (2 ∈ ℙ ∧ 𝑃 ∈ ℙ))
33		prmrp 16515	. . . . . . . . 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 256	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (2 gcd 𝑃) = 1)
36		odzphi 16595	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
37	24, 26, 35, 36	syl3anc 1370	. . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
38		phiprm 16576	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
39	38	ad2antll 726	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (ϕ‘𝑃) = (𝑃 − 1))
40	39	breq2d 5105	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃) ↔ ((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1)))
41		breq1 5096	. . . . . . . . . . 11 ⊢ (((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)) → (((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1) ↔ (2↑(𝑁 + 1)) ∥ (𝑃 − 1)))
42	41	adantl 482	. . . . . . . . . 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 12148	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
44	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
45		peano2nn 12087	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
46	45	nnnn0d 12395	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
47	44, 46	nnexpcld 14062	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
48	23	nnnn0d 12395	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
49		prmuz2 16499	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
50		eluzle 12697	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
51	49, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 2 ≤ 𝑃)
52		nn0ge2m1nn 12404	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℕ₀ ∧ 2 ≤ 𝑃) → (𝑃 − 1) ∈ ℕ)
53	48, 51, 52	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → (𝑃 − 1) ∈ ℕ)
54	47, 53	anim12i 613	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → ((2↑(𝑁 + 1)) ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ))
55	54	adantl 482	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((2↑(𝑁 + 1)) ∈ ℕ ∧ (𝑃 − 1) ∈ ℕ))
56		nndivides 16073	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . 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 12091	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℂ)
61	60	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → 𝑃 ∈ ℂ)
62	61	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 𝑃 ∈ ℂ)
63		1cnd 11072	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
64		nncn 12083	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
65	64	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
66		peano2nn0 12375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
67	3, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
68	44, 67	nnexpcld 14062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
69	68	nncnd 12091	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℂ)
70	69	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (2↑(𝑁 + 1)) ∈ ℂ)
71	70	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → (2↑(𝑁 + 1)) ∈ ℂ)
72	65, 71	mulcld 11097	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → (𝑘 · (2↑(𝑁 + 1))) ∈ ℂ)
73	62, 63, 72	subadd2d 11453	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → ((𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))) ↔ ((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃))
74	73	adantll 711	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → ((𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))) ↔ ((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃))
75		eqcom 2743	. . . . . . . . . . . . . . . 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 304	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) ∧ 𝑘 ∈ ℕ) → ((𝑘 · (2↑(𝑁 + 1))) = (𝑃 − 1) ↔ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
78	77	rexbidva 3169	. . . . . . . . . . . . 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 481	. . . . . . . . . 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 413	. . . . . . . 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 1170	. . . 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 413	. 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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3895 {csn 4574 class class class wbr 5093 ‘cfv 6480 (class class class)co 7338 ℂcc 10971 1c1 10974 + caddc 10976 · cmul 10978 ≤ cle 11112 − cmin 11307 ℕcn 12075 2c2 12130 ℕ₀cn0 12335 ℤcz 12421 ℤ_≥cuz 12684 ↑cexp 13884 ∥ cdvds 16063 gcd cgcd 16301 ℙcprime 16474 od_ℤcodz 16562 ϕcphi 16563 FermatNocfmtno 45397

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-pre-sup 11051

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-2o 8369 df-oadd 8372 df-er 8570 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-sup 9300 df-inf 9301 df-dju 9759 df-card 9797 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 df-nn 12076 df-2 12138 df-3 12139 df-n0 12336 df-xnn0 12408 df-z 12422 df-uz 12685 df-q 12791 df-rp 12833 df-fz 13342 df-fzo 13485 df-fl 13614 df-mod 13692 df-seq 13824 df-exp 13885 df-hash 14147 df-cj 14910 df-re 14911 df-im 14912 df-sqrt 15046 df-abs 15047 df-dvds 16064 df-gcd 16302 df-prm 16475 df-odz 16564 df-phi 16565 df-pc 16636 df-fmtno 45398

This theorem is referenced by: fmtnoprmfac2lem1 45436

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