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Theorem lgslem1 16119

Description: When 𝑎 is coprime to the prime 𝑝, 𝑎↑((𝑝 − 1) / 2) is equivalent mod 𝑝 to 1 or -1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.)

**Assertion**
Ref	Expression
lgslem1	⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})

**Proof of Theorem lgslem1**
Step	Hyp	Ref	Expression
1		eldifi 3351	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1050	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℙ)
3		prmnn 12888	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℕ)
5		simp1 1028	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℤ)
6		prmz 12889	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
7	2, 6	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℤ)
8	5, 7	gcdcomd 12751	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = (𝑃 gcd 𝐴))
9		simp3 1030	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ¬ 𝑃 ∥ 𝐴)
10		coprm 12922	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
11	2, 5, 10	syl2anc 415	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
12	9, 11	mpbid 147	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 gcd 𝐴) = 1)
13	8, 12	eqtrd 2271	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = 1)
14		eulerth 13011	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
15	4, 5, 13, 14	syl3anc 1278	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
16		phiprm 13001	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
17	2, 16	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) = (𝑃 − 1))
18		nnm1nn0 9604	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
19	4, 18	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 − 1) ∈ ℕ₀)
20	17, 19	eqeltrd 2315	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) ∈ ℕ₀)
21		zexpcl 10991	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑃) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
22	5, 20, 21	syl2anc 415	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
23		1zzd 9671	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 1 ∈ ℤ)
24		moddvds 12566	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ (𝐴↑(ϕ‘𝑃)) ∈ ℤ ∧ 1 ∈ ℤ) → (((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((𝐴↑(ϕ‘𝑃)) − 1)))
25	4, 22, 23, 24	syl3anc 1278	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃) ↔ 𝑃 ∥ ((𝐴↑(ϕ‘𝑃)) − 1)))
26	15, 25	mpbid 147	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∥ ((𝐴↑(ϕ‘𝑃)) − 1))
27	19	nn0cnd 9622	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 − 1) ∈ ℂ)
28		2cnd 9377	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℂ)
29		2ap0 9397	. . . . . . . . . . . . 13 ⊢ 2 # 0
30	29	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 # 0)
31	27, 28, 30	divcanap1d 9121	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
32	17, 31	eqtr4d 2274	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) = (((𝑃 − 1) / 2) · 2))
33	32	oveq2d 6101	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = (𝐴↑(((𝑃 − 1) / 2) · 2)))
34	5	zcnd 9769	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℂ)
35		2nn0 9580	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
36	35	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℕ₀)
37		oddprm 13038	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
38	37	3ad2ant2 1050	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ)
39	38	nnnn0d 9620	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ₀)
40	34, 36, 39	expmuld 11114	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(((𝑃 − 1) / 2) · 2)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
41	33, 40	eqtrd 2271	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
42	41	oveq1d 6100	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1))
43		sq1 11070	. . . . . . . 8 ⊢ (1↑2) = 1
44	43	oveq2i 6096	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1)
45	42, 44	eqtr4di 2289	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)))
46		zexpcl 10991	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
47	5, 39, 46	syl2anc 415	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
48	47	zcnd 9769	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℂ)
49		ax-1cn 8272	. . . . . . 7 ⊢ 1 ∈ ℂ
50		subsq 11083	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2)) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)) = (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
51	48, 49, 50	sylancl 417	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)) = (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
52	45, 51	eqtrd 2271	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
53	26, 52	breqtrd 4156	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
54	47	peano2zd 9771	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
55		peano2zm 9682	. . . . . 6 ⊢ ((𝐴↑((𝑃 − 1) / 2)) ∈ ℤ → ((𝐴↑((𝑃 − 1) / 2)) − 1) ∈ ℤ)
56	47, 55	syl 14	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) − 1) ∈ ℤ)
57		euclemma 12924	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ ∧ ((𝐴↑((𝑃 − 1) / 2)) − 1) ∈ ℤ) → (𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)) ↔ (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∨ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1))))
58	2, 54, 56, 57	syl3anc 1278	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)) ↔ (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∨ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1))))
59	53, 58	mpbid 147	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∨ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
60		dvdsval3 12558	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0))
61	4, 54, 60	syl2anc 415	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0))
62		2z 9672	. . . . . . 7 ⊢ 2 ∈ ℤ
63	62	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℤ)
64		moddvds 12566	. . . . . 6 ⊢ ((𝑃 ∈ ℕ ∧ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ ∧ 2 ∈ ℤ) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (2 mod 𝑃) ↔ 𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2)))
65	4, 54, 63, 64	syl3anc 1278	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (2 mod 𝑃) ↔ 𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2)))
66		zq 10026	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
67	62, 66	mp1i 10	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℚ)
68		zq 10026	. . . . . . . 8 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℚ)
69	7, 68	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℚ)
70		0le2 9394	. . . . . . . 8 ⊢ 0 ≤ 2
71	70	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 0 ≤ 2)
72		eldifsni 3843	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
73	72	3ad2ant2 1050	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ≠ 2)
74		zapne 9719	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑃 # 2 ↔ 𝑃 ≠ 2))
75	7, 62, 74	sylancl 417	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 # 2 ↔ 𝑃 ≠ 2))
76	73, 75	mpbird 167	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 # 2)
77		2re 9374	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
78	77	a1i 9	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℝ)
79	4	nnred 9317	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ)
80		prmuz2 12909	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
81	2, 80	syl 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ (ℤ_≥‘2))
82		eluzle 9934	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
83	81, 82	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ≤ 𝑃)
84	78, 79, 83	leltapd 8967	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (2 < 𝑃 ↔ 𝑃 # 2))
85	76, 84	mpbird 167	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 < 𝑃)
86		modqid 10786	. . . . . . 7 ⊢ (((2 ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
87	67, 69, 71, 85, 86	syl22anc 1279	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (2 mod 𝑃) = 2)
88	87	eqeq2d 2250	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = (2 mod 𝑃) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
89		df-2 9363	. . . . . . . 8 ⊢ 2 = (1 + 1)
90	89	oveq2i 6096	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) = (((𝐴↑((𝑃 − 1) / 2)) + 1) − (1 + 1))
91	49	a1i 9	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 1 ∈ ℂ)
92	48, 91, 91	pnpcan2d 8675	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − (1 + 1)) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
93	90, 92	eqtrid 2283	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
94	93	breq2d 4142	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
95	65, 88, 94	3bitr3rd 219	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
96	61, 95	orbi12d 805	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∨ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)) ↔ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2)))
97	59, 96	mpbid 147	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
98	54, 4	zmodcld 10782	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
99		elprg 3729	. . 3 ⊢ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀ → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2} ↔ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2)))
100	98, 99	syl 14	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2} ↔ ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2)))
101	97, 100	mpbird 167	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 720 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∖ cdif 3217 {csn 3709 {cpr 3710 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 ℝcr 8178 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 − cmin 8497 # cap 8909 / cdiv 9002 ℕcn 9304 2c2 9355 ℕ₀cn0 9563 ℤcz 9644 ℤ_≥cuz 9921 ℚcq 10019 mod cmo 10759 ↑cexp 10975 ∥ cdvds 12554 gcd cgcd 12730 ℙcprime 12885 ϕcphi 12987

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fz 10412 df-fzo 10550 df-fl 10705 df-mod 10760 df-seqfrec 10885 df-exp 10976 df-ihash 11215 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-clim 12045 df-proddc 12318 df-dvds 12555 df-gcd 12731 df-prm 12886 df-phi 12989

This theorem is used by: lgslem4 16122

W3C validator