Theorem odz2prm2pw 41475

Description: Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.)

Assertion

Ref	Expression
odz2prm2pw	⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))

Proof of Theorem odz2prm2pw

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
2		2nn 11185	. . . . . . . . 9 ⊢ 2 ∈ ℕ
3	2	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ)
4		2nn0 11309	. . . . . . . . . 10 ⊢ 2 ∈ ℕ0
5	4	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℕ0)
6		peano2nn 11032	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ)
7	6	nnnn0d 11351	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ0)
8	5, 7	nn0expcld 13031	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (2↑(𝑁 + 1)) ∈ ℕ0)
9	3, 8	nnexpcld 13030	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → (2↑(2↑(𝑁 + 1))) ∈ ℕ)
10	9	nnzd 11481	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑(2↑(𝑁 + 1))) ∈ ℤ)
11		modprm1div 15502	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (2↑(2↑(𝑁 + 1))) ∈ ℤ) → (((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1 ↔ 𝑃 ∥ ((2↑(2↑(𝑁 + 1))) − 1)))
12	1, 10, 11	syl2anr 495	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1 ↔ 𝑃 ∥ ((2↑(2↑(𝑁 + 1))) − 1)))
13		prmnn 15388	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
15	14	adantl 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑃 ∈ ℕ)
16		2z 11409	. . . . . . . 8 ⊢ 2 ∈ ℤ
17	16	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 2 ∈ ℤ)
18		eldifsn 4317	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ 𝑃 ≠ 2))
19		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 𝑃 ≠ 2)
20	19	necomd 2849	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ≠ 2) → 2 ≠ 𝑃)
21	18, 20	sylbi 207	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ≠ 𝑃)
22		2prm 15405	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
23		prmrp 15424	. . . . . . . . . 10 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((2 gcd 𝑃) = 1 ↔ 2 ≠ 𝑃))
24	22, 1, 23	sylancr 695	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 gcd 𝑃) = 1 ↔ 2 ≠ 𝑃))
25	21, 24	mpbird 247	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 gcd 𝑃) = 1)
26	25	adantl 482	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2 gcd 𝑃) = 1)
27	15, 17, 26	3jca 1242	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1))
28	8	adantr 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2↑(𝑁 + 1)) ∈ ℕ0)
29		odzdvds 15500	. . . . . 6 ⊢ (((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) ∧ (2↑(𝑁 + 1)) ∈ ℕ0) → (𝑃 ∥ ((2↑(2↑(𝑁 + 1))) − 1) ↔ ((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1))))
30	27, 28, 29	syl2anc 693	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃 ∥ ((2↑(2↑(𝑁 + 1))) − 1) ↔ ((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1))))
31	12, 30	bitrd 268	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1 ↔ ((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1))))
32		nnnn0 11299	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
33	5, 32	nn0expcld 13031	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ0)
34	3, 33	nnexpcld 13030	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (2↑(2↑𝑁)) ∈ ℕ)
35	34	nnzd 11481	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → (2↑(2↑𝑁)) ∈ ℤ)
36		modprm1div 15502	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ (2↑(2↑𝑁)) ∈ ℤ) → (((2↑(2↑𝑁)) mod 𝑃) = 1 ↔ 𝑃 ∥ ((2↑(2↑𝑁)) − 1)))
37	1, 35, 36	syl2anr 495	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑𝑁)) mod 𝑃) = 1 ↔ 𝑃 ∥ ((2↑(2↑𝑁)) − 1)))
38	33	adantr 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (2↑𝑁) ∈ ℕ0)
39		odzdvds 15500	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) ∧ (2↑𝑁) ∈ ℕ0) → (𝑃 ∥ ((2↑(2↑𝑁)) − 1) ↔ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁)))
40	27, 38, 39	syl2anc 693	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑃 ∥ ((2↑(2↑𝑁)) − 1) ↔ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁)))
41	37, 40	bitrd 268	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑𝑁)) mod 𝑃) = 1 ↔ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁)))
42	41	necon3abid 2830	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ↔ ¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁)))
43		odzcl 15498	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ (2 gcd 𝑃) = 1) → ((odℤ‘𝑃)‘2) ∈ ℕ)
44	27, 43	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((odℤ‘𝑃)‘2) ∈ ℕ)
45	7	adantr 481	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑁 + 1) ∈ ℕ0)
46		dvdsprmpweqle 15590	. . . . . . . . 9 ⊢ ((2 ∈ ℙ ∧ ((odℤ‘𝑃)‘2) ∈ ℕ ∧ (𝑁 + 1) ∈ ℕ0) → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → ∃𝑛 ∈ ℕ0 (𝑛 ≤ (𝑁 + 1) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛))))
47	22, 44, 45, 46	mp3an2i 1429	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → ∃𝑛 ∈ ℕ0 (𝑛 ≤ (𝑁 + 1) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛))))
48		breq1 4656	. . . . . . . . . . . . 13 ⊢ (((odℤ‘𝑃)‘2) = (2↑𝑛) → (((odℤ‘𝑃)‘2) ∥ (2↑𝑁) ↔ (2↑𝑛) ∥ (2↑𝑁)))
49	48	adantl 482	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (((odℤ‘𝑃)‘2) ∥ (2↑𝑁) ↔ (2↑𝑛) ∥ (2↑𝑁)))
50	49	notbid 308	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) ↔ ¬ (2↑𝑛) ∥ (2↑𝑁)))
51		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → ((odℤ‘𝑃)‘2) = (2↑𝑛))
52	51	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) ∧ ¬ (2↑𝑛) ∥ (2↑𝑁)) → ((odℤ‘𝑃)‘2) = (2↑𝑛))
53		nn0re 11301	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
54	6	nnred 11035	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℝ)
55	54	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝑁 + 1) ∈ ℝ)
56		leloe 10124	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℝ ∧ (𝑁 + 1) ∈ ℝ) → (𝑛 ≤ (𝑁 + 1) ↔ (𝑛 < (𝑁 + 1) ∨ 𝑛 = (𝑁 + 1))))
57	53, 55, 56	syl2anr 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑁 + 1) ↔ (𝑛 < (𝑁 + 1) ∨ 𝑛 = (𝑁 + 1))))
58		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
59		nn0z 11400	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
60	59	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℤ)
61	60	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → 𝑛 ∈ ℤ)
62		nnz 11399	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
63	62	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → 𝑁 ∈ ℤ)
64	63	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ ℤ)
65	64	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → 𝑁 ∈ ℤ)
66		zleltp1 11428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑛 ≤ 𝑁 ↔ 𝑛 < (𝑁 + 1)))
67	59, 63, 66	syl2anr 495	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ 𝑁 ↔ 𝑛 < (𝑁 + 1)))
68	67	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → 𝑛 ≤ 𝑁)
69		eluz2 11693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ (ℤ≥‘𝑛) ↔ (𝑛 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑛 ≤ 𝑁))
70	61, 65, 68, 69	syl3anbrc 1246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → 𝑁 ∈ (ℤ≥‘𝑛))
71		dvdsexp 15049	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((2 ∈ ℤ ∧ 𝑛 ∈ ℕ0 ∧ 𝑁 ∈ (ℤ≥‘𝑛)) → (2↑𝑛) ∥ (2↑𝑁))
72	16, 58, 70, 71	mp3an2ani 1431	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → (2↑𝑛) ∥ (2↑𝑁))
73	72	pm2.24d 147	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 < (𝑁 + 1)) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1))))
74	73	expcom 451	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 < (𝑁 + 1) → (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1)))))
75		oveq2 6658	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = (𝑁 + 1) → (2↑𝑛) = (2↑(𝑁 + 1)))
76	75	2a1d 26	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = (𝑁 + 1) → (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1)))))
77	74, 76	jaoi 394	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 < (𝑁 + 1) ∨ 𝑛 = (𝑁 + 1)) → (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1)))))
78	77	com12 32	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → ((𝑛 < (𝑁 + 1) ∨ 𝑛 = (𝑁 + 1)) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1)))))
79	57, 78	sylbid 230	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → (𝑛 ≤ (𝑁 + 1) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1)))))
80	79	imp 445	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1))))
81	80	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ (2↑𝑛) ∥ (2↑𝑁) → (2↑𝑛) = (2↑(𝑁 + 1))))
82	81	imp 445	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) ∧ ¬ (2↑𝑛) ∥ (2↑𝑁)) → (2↑𝑛) = (2↑(𝑁 + 1)))
83	52, 82	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) ∧ ¬ (2↑𝑛) ∥ (2↑𝑁)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
84	83	ex 450	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ (2↑𝑛) ∥ (2↑𝑁) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))
85	50, 84	sylbid 230	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 ≤ (𝑁 + 1)) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1))))
86	85	expl 648	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ≤ (𝑁 + 1) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
87	86	rexlimdva 3031	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (∃𝑛 ∈ ℕ0 (𝑛 ≤ (𝑁 + 1) ∧ ((odℤ‘𝑃)‘2) = (2↑𝑛)) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
88	47, 87	syld 47	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
89	88	com23 86	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (¬ ((odℤ‘𝑃)‘2) ∥ (2↑𝑁) → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
90	42, 89	sylbid 230	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
91	90	com23 86	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((odℤ‘𝑃)‘2) ∥ (2↑(𝑁 + 1)) → (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
92	31, 91	sylbid 230	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1 → (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
93	92	com23 86	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 → (((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1 → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))))
94	93	imp32 449	1 ⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∖ cdif 3571  {csn 4177   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687   mod cmo 12668  ↑cexp 12860   ∥ cdvds 14983   gcd cgcd 15216  ℙcprime 15385  od_ℤcodz 15468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-odz 15470  df-phi 15471  df-pc 15542

This theorem is referenced by:  fmtnoprmfac1lem  41476