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Theorem m1lgs 15242

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 9352	. . . . . . . . 9 ⊢ -1 ∈ ℤ
2		oddprm 12400	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	nnnn0d 9296	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ₀)
4		zexpcl 10628	. . . . . . . . 9 ⊢ ((-1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
5	1, 3, 4	sylancr 414	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
6	5	peano2zd 9445	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
7		eldifi 3282	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 12251	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	7, 8	syl 14	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
10	6, 9	zmodcld 10419	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
11	10	nn0cnd 9298	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℂ)
12		1cnd 8037	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℂ)
13	11, 12, 12	subaddd 8350	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
14		2z 9348	. . . . . . . . 9 ⊢ 2 ∈ ℤ
15		zq 9694	. . . . . . . . 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 12252	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
19		zq 9694	. . . . . . . 8 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ ℚ)
20	7, 18, 19	3syl 17	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℚ)
21		0le2 9074	. . . . . . . 8 ⊢ 0 ≤ 2
22	21	a1i 9	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 0 ≤ 2)
23		oddprmgt2 12275	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
24		modqid 10423	. . . . . . 7 ⊢ (((2 ∈ ℚ ∧ 𝑃 ∈ ℚ) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
25	17, 20, 22, 23, 24	syl22anc 1250	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = 2)
26		df-2 9043	. . . . . 6 ⊢ 2 = (1 + 1)
27	25, 26	eqtrdi 2242	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = (1 + 1))
28	27	eqeq1d 2202	. . . 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 9146	. . . . . . . 8 ⊢ 2 ∈ ℕ
30	2	nnzd 9441	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ)
31		dvdsdc 11944	. . . . . . . 8 ⊢ ((2 ∈ ℕ ∧ ((𝑃 − 1) / 2) ∈ ℤ) → DECID 2 ∥ ((𝑃 − 1) / 2))
32	29, 30, 31	sylancr 414	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → DECID 2 ∥ ((𝑃 − 1) / 2))
33		eldifsni 3748	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
34	33	neneqd 2385	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 = 2)
35		prmuz2 12272	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
36	7, 35	syl 14	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
37		2prm 12268	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
38		dvdsprm 12278	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
39	36, 37, 38	sylancl 413	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
40	34, 39	mtbird 674	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 ∥ 2)
41	40	adantr 276	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ 2)
42		1cnd 8037	. . . . . . . . . . . . . . . 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 12021	. . . . . . . . . . . . . . . 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 1249	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
47	43	nnzd 9441	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℤ)
48		1exp 10642	. . . . . . . . . . . . . . . . 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 8216	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1)
51	46, 50	eqtrd 2226	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1)
52	51	oveq1d 5934	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 + 1))
53		ax-1cn 7967	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
54		neg1cn 9089	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
55		1pneg1e0 9095	. . . . . . . . . . . . . 14 ⊢ (1 + -1) = 0
56	53, 54, 55	addcomli 8166	. . . . . . . . . . . . 13 ⊢ (-1 + 1) = 0
57	52, 56	eqtrdi 2242	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0)
58	57	oveq2d 5935	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 − 0))
59		2cn 9055	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
60	59	subid1i 8293	. . . . . . . . . . 11 ⊢ (2 − 0) = 2
61	58, 60	eqtrdi 2242	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = 2)
62	61	breq2d 4042	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2))
63	41, 62	mtbird 674	. . . . . . . 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 854	. . . . . . 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 11945	. . . . . . 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 1249	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
70		4z 9350	. . . . . . . . 9 ⊢ 4 ∈ ℤ
71		4ne0 9082	. . . . . . . . 9 ⊢ 4 ≠ 0
72		nnm1nn0 9284	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
73	9, 72	syl 14	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℕ₀)
74	73	nn0zd 9440	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℤ)
75		dvdsval2 11936	. . . . . . . . 9 ⊢ ((4 ∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
76	70, 71, 74, 75	mp3an12i 1352	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
77	73	nn0cnd 9298	. . . . . . . . . . 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 9030	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 # 0)
81	77, 78, 78, 80, 80	divdivap1d 8843	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / (2 · 2)))
82		2t2e4 9139	. . . . . . . . . . 11 ⊢ (2 · 2) = 4
83	82	oveq2i 5930	. . . . . . . . . 10 ⊢ ((𝑃 − 1) / (2 · 2)) = ((𝑃 − 1) / 4)
84	81, 83	eqtrdi 2242	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / 4))
85	84	eleq1d 2262	. . . . . . . 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 9076	. . . . . . . 8 ⊢ 2 ≠ 0
88		dvdsval2 11936	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
89	14, 87, 30, 88	mp3an12i 1352	. . . . . . 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 9093	. . . . . . . . . . . 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 10659	. . . . . . . . . . 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 1250	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
99	2	nncnd 8998	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℂ)
100	99, 78, 80	divcanap2d 8813	. . . . . . . . . . . 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 5935	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) / 2)))
103		neg1sqe1 10708	. . . . . . . . . . . 12 ⊢ (-1↑2) = 1
104	103	oveq1i 5929	. . . . . . . . . . 11 ⊢ ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) / 2))
105		1exp 10642	. . . . . . . . . . . 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 2238	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1)
108	98, 102, 107	3eqtr3d 2234	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑((𝑃 − 1) / 2)) = 1)
109	108	oveq1d 5934	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (1 + 1))
110	26, 109	eqtr4id 2245	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1))
111	110	oveq1d 5934	. . . . . 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 15175	. . . . 5 ⊢ ((-1 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
116	1, 115	mpan 424	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
117	116	eqeq1d 2202	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1))
118		4nn 9148	. . . . 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 9347	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℤ)
122		moddvds 11945	. . . 4 ⊢ ((4 ∈ ℕ ∧ 𝑃 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
123	119, 120, 121, 122	syl3anc 1249	. . 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 9346	. . . . 5 ⊢ 1 ∈ ℤ
126		zq 9694	. . . . 5 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
127	125, 126	ax-mp 5	. . . 4 ⊢ 1 ∈ ℚ
128		zq 9694	. . . . 5 ⊢ (4 ∈ ℤ → 4 ∈ ℚ)
129	70, 128	ax-mp 5	. . . 4 ⊢ 4 ∈ ℚ
130		0le1 8502	. . . 4 ⊢ 0 ≤ 1
131		1lt4 9159	. . . 4 ⊢ 1 < 4
132		modqid 10423	. . . 4 ⊢ (((1 ∈ ℚ ∧ 4 ∈ ℚ) ∧ (0 ≤ 1 ∧ 1 < 4)) → (1 mod 4) = 1)
133	127, 129, 130, 131, 132	mp4an 427	. . 3 ⊢ (1 mod 4) = 1
134	133	eqeq2i 2204	. 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

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ∖ cdif 3151 {csn 3619 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 ℂcc 7872 0cc0 7874 1c1 7875 + caddc 7877 · cmul 7879 < clt 8056 ≤ cle 8057 − cmin 8192 -cneg 8193 # cap 8602 / cdiv 8693 ℕcn 8984 2c2 9035 4c4 9037 ℕ₀cn0 9243 ℤcz 9320 ℤ_≥cuz 9595 ℚcq 9687 mod cmo 10396 ↑cexp 10612 ∥ cdvds 11933 ℙcprime 12248 /_L clgs 15154

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-2o 6472 df-oadd 6475 df-er 6589 df-en 6797 df-dom 6798 df-fin 6799 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-proddc 11697 df-dvds 11934 df-gcd 12083 df-prm 12249 df-phi 12352 df-pc 12426 df-lgs 15155

This theorem is referenced by: (None)

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