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Theorem lgslem1 16003

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 3345	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1046	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℙ)
3		prmnn 12836	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℕ)
5		simp1 1024	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℤ)
6		prmz 12837	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
7	2, 6	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℤ)
8	5, 7	gcdcomd 12699	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = (𝑃 gcd 𝐴))
9		simp3 1026	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ¬ 𝑃 ∥ 𝐴)
10		coprm 12870	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
11	2, 5, 10	syl2anc 411	. . . . . . . . 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 2267	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = 1)
14		eulerth 12959	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
15	4, 5, 13, 14	syl3anc 1274	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
16		phiprm 12949	. . . . . . . . . 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 9557	. . . . . . . . . 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 2311	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) ∈ ℕ₀)
21		zexpcl 10943	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑃) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
22	5, 20, 21	syl2anc 411	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
23		1zzd 9624	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 1 ∈ ℤ)
24		moddvds 12514	. . . . . . 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 1274	. . . . . 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 9575	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 − 1) ∈ ℂ)
28		2cnd 9330	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℂ)
29		2ap0 9350	. . . . . . . . . . . . 13 ⊢ 2 # 0
30	29	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 # 0)
31	27, 28, 30	divcanap1d 9085	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
32	17, 31	eqtr4d 2270	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) = (((𝑃 − 1) / 2) · 2))
33	32	oveq2d 6074	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = (𝐴↑(((𝑃 − 1) / 2) · 2)))
34	5	zcnd 9722	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℂ)
35		2nn0 9533	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
36	35	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℕ₀)
37		oddprm 12986	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
38	37	3ad2ant2 1046	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ)
39	38	nnnn0d 9573	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ₀)
40	34, 36, 39	expmuld 11066	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(((𝑃 − 1) / 2) · 2)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
41	33, 40	eqtrd 2267	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
42	41	oveq1d 6073	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1))
43		sq1 11022	. . . . . . . 8 ⊢ (1↑2) = 1
44	43	oveq2i 6069	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1)
45	42, 44	eqtr4di 2285	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)))
46		zexpcl 10943	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
47	5, 39, 46	syl2anc 411	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
48	47	zcnd 9722	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℂ)
49		ax-1cn 8236	. . . . . . 7 ⊢ 1 ∈ ℂ
50		subsq 11035	. . . . . . 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 413	. . . . . 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 2267	. . . . 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 4140	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
54	47	peano2zd 9724	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
55		peano2zm 9635	. . . . . 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 12872	. . . . 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 1274	. . . 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 12506	. . . . 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 411	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0))
62		2z 9625	. . . . . . 7 ⊢ 2 ∈ ℤ
63	62	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℤ)
64		moddvds 12514	. . . . . 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 1274	. . . . 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 9979	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
67	62, 66	mp1i 10	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℚ)
68		zq 9979	. . . . . . . 8 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℚ)
69	7, 68	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℚ)
70		0le2 9347	. . . . . . . 8 ⊢ 0 ≤ 2
71	70	a1i 9	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 0 ≤ 2)
72		eldifsni 3827	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
73	72	3ad2ant2 1046	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ≠ 2)
74		zapne 9672	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℤ) → (𝑃 # 2 ↔ 𝑃 ≠ 2))
75	7, 62, 74	sylancl 413	. . . . . . . . 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 9327	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
78	77	a1i 9	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℝ)
79	4	nnred 9270	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ)
80		prmuz2 12857	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
81	2, 80	syl 14	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ (ℤ_≥‘2))
82		eluzle 9887	. . . . . . . . . 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 8931	. . . . . . . 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 10738	. . . . . . 7 ⊢ (((2 ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
87	67, 69, 71, 85, 86	syl22anc 1275	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (2 mod 𝑃) = 2)
88	87	eqeq2d 2246	. . . . 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 9316	. . . . . . . 8 ⊢ 2 = (1 + 1)
90	89	oveq2i 6069	. . . . . . 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 8639	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − (1 + 1)) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
93	90, 92	eqtrid 2279	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
94	93	breq2d 4126	. . . . 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 801	. . 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 10734	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
99		elprg 3714	. . 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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∖ cdif 3211 {csn 3694 {cpr 3695 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 1c1 8144 + caddc 8146 · cmul 8148 < clt 8324 ≤ cle 8325 − cmin 8461 # cap 8873 / cdiv 8966 ℕcn 9257 2c2 9308 ℕ₀cn0 9516 ℤcz 9597 ℤ_≥cuz 9874 ℚcq 9972 mod cmo 10711 ↑cexp 10927 ∥ cdvds 12502 gcd cgcd 12678 ℙcprime 12833 ϕcphi 12935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8463 df-neg 8464 df-reap 8867 df-ap 8874 df-div 8967 df-inn 9258 df-2 9316 df-3 9317 df-4 9318 df-n0 9517 df-z 9598 df-uz 9875 df-q 9973 df-rp 10008 df-fz 10365 df-fzo 10502 df-fl 10657 df-mod 10712 df-seqfrec 10837 df-exp 10928 df-ihash 11167 df-cj 11555 df-re 11556 df-im 11557 df-rsqrt 11712 df-abs 11713 df-clim 11993 df-proddc 12266 df-dvds 12503 df-gcd 12679 df-prm 12834 df-phi 12937

This theorem is referenced by: lgslem4 16006

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