fmtnoprmfac1 - Mathbox for Alexander van der Vekens

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Theorem fmtnoprmfac1 45747

Description: Divisor of Fermat number (special form of Euler's result, see fmtnofac1 45752): 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 5108	. . . . . . 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 12420	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
4		fmtnoodd 45715	. . . . . . . . 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 1348	. 2 ⊢ (𝑃 = 2 → ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1)))
12		simpr1 1194	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑁 ∈ ℕ)
13		neqne 2951	. . . . . . . . . 10 ⊢ (¬ 𝑃 = 2 → 𝑃 ≠ 2)
14	13	anim2i 617	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2) → (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
15		eldifsn 4747	. . . . . . . . 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 1134	. . . . . 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 1196	. . . . 5 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → 𝑃 ∥ (FermatNo‘𝑁))
21		fmtnoprmfac1lem 45746	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
22	12, 19, 20, 21	syl3anc 1371	. . . 4 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁))) → ((od_ℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
23		prmnn 16550	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
24	23	ad2antll 727	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 𝑃 ∈ ℕ)
25		2z 12535	. . . . . . . 8 ⊢ 2 ∈ ℤ
26	25	a1i 11	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ∈ ℤ)
27	13	necomd 2999	. . . . . . . . 9 ⊢ (¬ 𝑃 = 2 → 2 ≠ 𝑃)
28	27	adantr 481	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → 2 ≠ 𝑃)
29		2prm 16568	. . . . . . . . . . . 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 16588	. . . . . . . . 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 16668	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
37	24, 26, 35, 36	syl3anc 1371	. . . . . 6 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → ((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃))
38		phiprm 16649	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
39	38	ad2antll 727	. . . . . . . 8 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (ϕ‘𝑃) = (𝑃 − 1))
40	39	breq2d 5117	. . . . . . 7 ⊢ ((¬ 𝑃 = 2 ∧ (𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ)) → (((od_ℤ‘𝑃)‘2) ∥ (ϕ‘𝑃) ↔ ((od_ℤ‘𝑃)‘2) ∥ (𝑃 − 1)))
41		breq1 5108	. . . . . . . . . . 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 12226	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℕ
44	43	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
45		peano2nn 12165	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
46	45	nnnn0d 12473	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
47	44, 46	nnexpcld 14148	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
48	23	nnnn0d 12473	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
49		prmuz2 16572	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
50		eluzle 12776	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
51	49, 50	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ ℙ → 2 ≤ 𝑃)
52		nn0ge2m1nn 12482	. . . . . . . . . . . . . . . 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 16146	. . . . . . . . . . . . 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 12169	. . . . . . . . . . . . . . . . . . 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 11150	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
64		nncn 12161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
65	64	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
66		peano2nn0 12453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ₀)
67	3, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ₀)
68	44, 67	nnexpcld 14148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ)
69	68	nncnd 12169	. . . . . . . . . . . . . . . . . . . 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 11175	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → (𝑘 · (2↑(𝑁 + 1))) ∈ ℂ)
73	62, 63, 72	subadd2d 11531	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) ∧ 𝑘 ∈ ℕ) → ((𝑃 − 1) = (𝑘 · (2↑(𝑁 + 1))) ↔ ((𝑘 · (2↑(𝑁 + 1))) + 1) = 𝑃))
74	73	adantll 712	. . . . . . . . . . . . . . 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 3173	. . . . . . . . . . . . 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 1171	. . . 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 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ∖ cdif 3907 {csn 4586 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 1c1 11052 + caddc 11054 · cmul 11056 ≤ cle 11190 − cmin 11385 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ↑cexp 13967 ∥ cdvds 16136 gcd cgcd 16374 ℙcprime 16547 od_ℤcodz 16635 ϕcphi 16636 FermatNocfmtno 45709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-prm 16548 df-odz 16637 df-phi 16638 df-pc 16709 df-fmtno 45710

This theorem is referenced by: fmtnoprmfac2lem1 45748

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