m1lgs - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > m1lgs		GIF version

Theorem m1lgs 16204

Description: The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime 𝑃 iff 𝑃≡1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)

**Assertion**
Ref	Expression
m1lgs	⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = 1))

**Proof of Theorem m1lgs**
Step	Hyp	Ref	Expression
1		neg1z 9676	. . . . . . . . 9 ⊢ -1 ∈ ℤ
2		oddprm 13038	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	nnnn0d 9620	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ₀)
4		zexpcl 10991	. . . . . . . . 9 ⊢ ((-1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
5	1, 3, 4	sylancr 418	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
6	5	peano2zd 9771	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
7		eldifi 3351	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 12888	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
10	6, 9	zmodcld 10782	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
11	10	nn0cnd 9622	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℂ)
12		1cnd 8342	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℂ)
13	11, 12, 12	subaddd 8655	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
14		2z 9672	. . . . . . . . 9 ⊢ 2 ∈ ℤ
15		zq 10026	. . . . . . . . 9 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
16	14, 15	ax-mp 5	. . . . . . . 8 ⊢ 2 ∈ ℚ
17	16	a1i 9	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℚ)
18		prmz 12889	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
19		zq 10026	. . . . . . . 8 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℚ)
20	7, 18, 19	3syl 17	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℚ)
21		0le2 9394	. . . . . . . 8 ⊢ 0 ≤ 2
22	21	a1i 9	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 0 ≤ 2)
23		oddprmgt2 12912	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
24		modqid 10786	. . . . . . 7 ⊢ (((2 ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
25	17, 20, 22, 23, 24	syl22anc 1279	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = 2)
26		df-2 9363	. . . . . 6 ⊢ 2 = (1 + 1)
27	25, 26	eqtrdi 2287	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = (1 + 1))
28	27	eqeq1d 2247	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
29		2nn 9466	. . . . . . . 8 ⊢ 2 ∈ ℕ
30	2	nnzd 9767	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ)
31		dvdsdc 12565	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℤ) → DECID 2 ∥ ((𝑃 − 1) / 2))
32	29, 30, 31	sylancr 418	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → DECID 2 ∥ ((𝑃 − 1) / 2))
33		eldifsni 3843	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
34	33	neneqd 2441	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 = 2)
35		prmuz2 12909	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
36	7, 35	syl 14	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
37		2prm 12905	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
38		dvdsprm 12915	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
39	36, 37, 38	sylancl 417	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
40	34, 39	mtbird 684	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 ∥ 2)
41	40	adantr 276	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ 2)
42		1cnd 8342	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → 1 ∈ ℂ)
43	2	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℕ)
44		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2))
45		oexpneg 12644	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
46	42, 43, 44, 45	syl3anc 1278	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
47	43	nnzd 9767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℤ)
48		1exp 11005	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → (1↑((𝑃 − 1) / 2)) = 1)
49	47, 48	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (1↑((𝑃 − 1) / 2)) = 1)
50	49	negeqd 8521	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1)
51	46, 50	eqtrd 2271	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1)
52	51	oveq1d 6100	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 + 1))
53		ax-1cn 8272	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
54		neg1cn 9409	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
55		1pneg1e0 9415	. . . . . . . . . . . . . 14 ⊢ (1 + -1) = 0
56	53, 54, 55	addcomli 8471	. . . . . . . . . . . . 13 ⊢ (-1 + 1) = 0
57	52, 56	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0)
58	57	oveq2d 6101	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 − 0))
59		2cn 9375	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
60	59	subid1i 8598	. . . . . . . . . . 11 ⊢ (2 − 0) = 2
61	58, 60	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = 2)
62	61	breq2d 4142	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2))
63	41, 62	mtbird 684	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)))
64	63	ex 115	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (¬ 2 ∥ ((𝑃 − 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
65		condc 865	. . . . . . 7 ⊢ (DECID 2 ∥ ((𝑃 − 1) / 2) → ((¬ 2 ∥ ((𝑃 − 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2))))
66	32, 64, 65	sylc 62	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2)))
67	14	a1i 9	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℤ)
68		moddvds 12566	. . . . . . 7 ⊢ ((𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
69	9, 67, 6, 68	syl3anc 1278	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
70		4z 9674	. . . . . . . . 9 ⊢ 4 ∈ ℤ
71		4ne0 9402	. . . . . . . . 9 ⊢ 4 ≠ 0
72		nnm1nn0 9604	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
73	9, 72	syl 14	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℕ₀)
74	73	nn0zd 9766	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℤ)
75		dvdsval2 12557	. . . . . . . . 9 ⊢ ((4 ∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
76	70, 71, 74, 75	mp3an12i 1382	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
77	73	nn0cnd 9622	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℂ)
78	59	a1i 9	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℂ)
79	29	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℕ)
80	79	nnap0d 9350	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 # 0)
81	77, 78, 78, 80, 80	divdivap1d 9152	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / (2 · 2)))
82		2t2e4 9459	. . . . . . . . . . 11 ⊢ (2 · 2) = 4
83	82	oveq2i 6096	. . . . . . . . . 10 ⊢ ((𝑃 − 1) / (2 · 2)) = ((𝑃 − 1) / 4)
84	81, 83	eqtrdi 2287	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / 4))
85	84	eleq1d 2307	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((((𝑃 − 1) / 2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈ ℤ))
86	76, 85	bitr4d 191	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
87		2ne0 9396	. . . . . . . 8 ⊢ 2 ≠ 0
88		dvdsval2 12557	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
89	14, 87, 30, 88	mp3an12i 1382	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
90	86, 89	bitr4d 191	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ 2 ∥ ((𝑃 − 1) / 2)))
91	66, 69, 90	3imtr4d 203	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) → 4 ∥ (𝑃 − 1)))
92	54	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → -1 ∈ ℂ)
93		neg1ap0 9413	. . . . . . . . . . . 12 ⊢ -1 # 0
94	93	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → -1 # 0)
95	14	a1i 9	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 ∈ ℤ)
96	86	biimpa 296	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (((𝑃 − 1) / 2) / 2) ∈ ℤ)
97		expmulzap 11022	. . . . . . . . . . 11 ⊢ (((-1 ∈ ℂ ∧ -1 # 0) ∧ (2 ∈ ℤ ∧ (((𝑃 − 1) / 2) / 2) ∈ ℤ)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
98	92, 94, 95, 96, 97	syl22anc 1279	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
99	2	nncnd 9318	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℂ)
100	99, 78, 80	divcanap2d 9122	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
101	100	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
102	101	oveq2d 6101	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) / 2)))
103		neg1sqe1 11071	. . . . . . . . . . . 12 ⊢ (-1↑2) = 1
104	103	oveq1i 6095	. . . . . . . . . . 11 ⊢ ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) / 2))
105		1exp 11005	. . . . . . . . . . . 12 ⊢ ((((𝑃 − 1) / 2) / 2) ∈ ℤ → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
106	96, 105	syl 14	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
107	104, 106	eqtrid 2283	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1)
108	98, 102, 107	3eqtr3d 2279	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑((𝑃 − 1) / 2)) = 1)
109	108	oveq1d 6100	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (1 + 1))
110	26, 109	eqtr4id 2290	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1))
111	110	oveq1d 6100	. . . . . 6 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃))
112	111	ex 115	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
113	91, 112	impbid 129	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 4 ∥ (𝑃 − 1)))
114	13, 28, 113	3bitr2d 216	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1)))
115		lgsval3 16137	. . . . 5 ⊢ ((-1 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
116	1, 115	mpan 428	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
117	116	eqeq1d 2247	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1))
118		4nn 9468	. . . . 5 ⊢ 4 ∈ ℕ
119	118	a1i 9	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 4 ∈ ℕ)
120	7, 18	syl 14	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
121		1zzd 9671	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℤ)
122		moddvds 12566	. . . 4 ⊢ ((4 ∈ ℕ ∧ 𝑃 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
123	119, 120, 121, 122	syl3anc 1278	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
124	114, 117, 123	3bitr4d 220	. 2 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4)))
125		1z 9670	. . . . 5 ⊢ 1 ∈ ℤ
126		zq 10026	. . . . 5 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
127	125, 126	ax-mp 5	. . . 4 ⊢ 1 ∈ ℚ
128		zq 10026	. . . . 5 ⊢ (4 ∈ ℤ → 4 ∈ ℚ)
129	70, 128	ax-mp 5	. . . 4 ⊢ 4 ∈ ℚ
130		0le1 8809	. . . 4 ⊢ 0 ≤ 1
131		1lt4 9479	. . . 4 ⊢ 1 < 4
132		modqid 10786	. . . 4 ⊢ (((1 ∈ ℚ ∧ 4 ∈ ℚ) ∧ (0 ≤ 1 ∧ 1 < 4)) → (1 mod 4) = 1)
133	127, 129, 130, 131, 132	mp4an 431	. . 3 ⊢ (1 mod 4) = 1
134	133	eqeq2i 2249	. 2 ⊢ ((𝑃 mod 4) = (1 mod 4) ↔ (𝑃 mod 4) = 1)
135	124, 134	bitrdi 196	1 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = 1))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∖ cdif 3217 {csn 3709 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 − cmin 8497 -cneg 8498 # cap 8909 / cdiv 9002 ℕcn 9304 2c2 9355 4c4 9357 ℕ₀cn0 9563 ℤcz 9644 ℤ_≥cuz 9921 ℚcq 10019 mod cmo 10759 ↑cexp 10975 ∥ cdvds 12554 ℙcprime 12885 /_L clgs 16116

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-inf 7325 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-5 9366 df-6 9367 df-7 9368 df-8 9369 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 df-pc 13064 df-lgs 16117

This theorem is used by: (None)

W3C validator