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Theorem lgslem1 26033

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 4017	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
2	1	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℙ)
3		prmnn 16115	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℕ)
5		simp1 1137	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℤ)
6		prmz 16116	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
7	2, 6	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℤ)
8	5, 7	gcdcomd 15957	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = (𝑃 gcd 𝐴))
9		simp3 1139	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ¬ 𝑃 ∥ 𝐴)
10		coprm 16152	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
11	2, 5, 10	syl2anc 587	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (¬ 𝑃 ∥ 𝐴 ↔ (𝑃 gcd 𝐴) = 1))
12	9, 11	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 gcd 𝐴) = 1)
13	8, 12	eqtrd 2773	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴 gcd 𝑃) = 1)
14		eulerth 16220	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ (𝐴 gcd 𝑃) = 1) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
15	4, 5, 13, 14	syl3anc 1372	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) mod 𝑃) = (1 mod 𝑃))
16		phiprm 16214	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → (ϕ‘𝑃) = (𝑃 − 1))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) = (𝑃 − 1))
18		nnm1nn0 12017	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
19	4, 18	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 − 1) ∈ ℕ₀)
20	17, 19	eqeltrd 2833	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) ∈ ℕ₀)
21		zexpcl 13536	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ (ϕ‘𝑃) ∈ ℕ₀) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
22	5, 20, 21	syl2anc 587	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) ∈ ℤ)
23		1zzd 12094	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 1 ∈ ℤ)
24		moddvds 15710	. . . . . . 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 1372	. . . . . 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 235	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∥ ((𝐴↑(ϕ‘𝑃)) − 1))
27	19	nn0cnd 12038	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 − 1) ∈ ℂ)
28		2cnd 11794	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℂ)
29		2ne0 11820	. . . . . . . . . . . . 13 ⊢ 2 ≠ 0
30	29	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ≠ 0)
31	27, 28, 30	divcan1d 11495	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝑃 − 1) / 2) · 2) = (𝑃 − 1))
32	17, 31	eqtr4d 2776	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (ϕ‘𝑃) = (((𝑃 − 1) / 2) · 2))
33	32	oveq2d 7186	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = (𝐴↑(((𝑃 − 1) / 2) · 2)))
34	5	zcnd 12169	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝐴 ∈ ℂ)
35		2nn0 11993	. . . . . . . . . . 11 ⊢ 2 ∈ ℕ₀
36	35	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℕ₀)
37		oddprm 16247	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
38	37	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ)
39	38	nnnn0d 12036	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝑃 − 1) / 2) ∈ ℕ₀)
40	34, 36, 39	expmuld 13605	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(((𝑃 − 1) / 2) · 2)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
41	33, 40	eqtrd 2773	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑(ϕ‘𝑃)) = ((𝐴↑((𝑃 − 1) / 2))↑2))
42	41	oveq1d 7185	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1))
43		sq1 13650	. . . . . . . 8 ⊢ (1↑2) = 1
44	43	oveq2i 7181	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)) = (((𝐴↑((𝑃 − 1) / 2))↑2) − 1)
45	42, 44	eqtr4di 2791	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑(ϕ‘𝑃)) − 1) = (((𝐴↑((𝑃 − 1) / 2))↑2) − (1↑2)))
46		zexpcl 13536	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
47	5, 39, 46	syl2anc 587	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℤ)
48	47	zcnd 12169	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝐴↑((𝑃 − 1) / 2)) ∈ ℂ)
49		ax-1cn 10673	. . . . . . 7 ⊢ 1 ∈ ℂ
50		subsq 13664	. . . . . . 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 589	. . . . . 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 2773	. . . . 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 5056	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) · ((𝐴↑((𝑃 − 1) / 2)) − 1)))
54	47	peano2zd 12171	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
55		peano2zm 12106	. . . . . 6 ⊢ ((𝐴↑((𝑃 − 1) / 2)) ∈ ℤ → ((𝐴↑((𝑃 − 1) / 2)) − 1) ∈ ℤ)
56	47, 55	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((𝐴↑((𝑃 − 1) / 2)) − 1) ∈ ℤ)
57		euclemma 16154	. . . . 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 1372	. . . 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 235	. . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ∨ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
60		dvdsval3 15703	. . . . 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 587	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) + 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0))
62		2z 12095	. . . . . . 7 ⊢ 2 ∈ ℤ
63	62	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℤ)
64		moddvds 15710	. . . . . 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 1372	. . . . 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		2re 11790	. . . . . . . 8 ⊢ 2 ∈ ℝ
67	66	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ∈ ℝ)
68	4	nnrpd 12512	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ⁺)
69		0le2 11818	. . . . . . . 8 ⊢ 0 ≤ 2
70	69	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 0 ≤ 2)
71	4	nnred 11731	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ ℝ)
72		prmuz2 16137	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
73	2, 72	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ∈ (ℤ_≥‘2))
74		eluzle 12337	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℤ_≥‘2) → 2 ≤ 𝑃)
75	73, 74	syl 17	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 ≤ 𝑃)
76		eldifsni 4678	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
77	76	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 𝑃 ≠ 2)
78	67, 71, 75, 77	leneltd 10872	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 2 < 𝑃)
79		modid 13355	. . . . . . 7 ⊢ (((2 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
80	67, 68, 70, 78, 79	syl22anc 838	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (2 mod 𝑃) = 2)
81	80	eqeq2d 2749	. . . . 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))
82		df-2 11779	. . . . . . . 8 ⊢ 2 = (1 + 1)
83	82	oveq2i 7181	. . . . . . 7 ⊢ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) = (((𝐴↑((𝑃 − 1) / 2)) + 1) − (1 + 1))
84	49	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → 1 ∈ ℂ)
85	48, 84, 84	pnpcan2d 11113	. . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − (1 + 1)) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
86	83, 85	syl5eq 2785	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) = ((𝐴↑((𝑃 − 1) / 2)) − 1))
87	86	breq2d 5042	. . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ (((𝐴↑((𝑃 − 1) / 2)) + 1) − 2) ↔ 𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1)))
88	65, 81, 87	3bitr3rd 313	. . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (𝑃 ∥ ((𝐴↑((𝑃 − 1) / 2)) − 1) ↔ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
89	61, 88	orbi12d 918	. . 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)))
90	59, 89	mpbid 235	. 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 0 ∨ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) = 2))
91		ovex 7203	. . 3 ⊢ (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ V
92	91	elpr 4539	. 2 ⊢ ((((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))
93	90, 92	sylibr 237	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2})

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∖ cdif 3840 {csn 4516 {cpr 4518 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 + caddc 10618 · cmul 10620 < clt 10753 ≤ cle 10754 − cmin 10948 / cdiv 11375 ℕcn 11716 2c2 11771 ℕ₀cn0 11976 ℤcz 12062 ℤ_≥cuz 12324 ℝ⁺crp 12472 mod cmo 13328 ↑cexp 13521 ∥ cdvds 15699 gcd cgcd 15937 ℙcprime 16112 ϕcphi 16201

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-oadd 8135 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-xnn0 12049 df-z 12063 df-uz 12325 df-rp 12473 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-dvds 15700 df-gcd 15938 df-prm 16113 df-phi 16203

This theorem is referenced by: lgslem4 26036

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