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Theorem m1lgs 27326

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 12508	. . . . . . . . 9 ⊢ -1 ∈ ℤ
2		oddprm 16722	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	nnnn0d 12442	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ₀)
4		zexpcl 13983	. . . . . . . . 9 ⊢ ((-1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
5	1, 3, 4	sylancr 587	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
6	5	peano2zd 12580	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
7		eldifi 4078	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16585	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
10	6, 9	zmodcld 13796	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
11	10	nn0cnd 12444	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℂ)
12		1cnd 11107	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℂ)
13	11, 12, 12	subaddd 11490	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
14		2re 12199	. . . . . . . 8 ⊢ 2 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℝ)
16	9	nnrpd 12932	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ⁺)
17		0le2 12227	. . . . . . . 8 ⊢ 0 ≤ 2
18	17	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 0 ≤ 2)
19		oddprmgt2 16610	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
20		modid 13800	. . . . . . 7 ⊢ (((2 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
21	15, 16, 18, 19, 20	syl22anc 838	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = 2)
22		df-2 12188	. . . . . 6 ⊢ 2 = (1 + 1)
23	21, 22	eqtrdi 2782	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = (1 + 1))
24	23	eqeq1d 2733	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
25		eldifsni 4739	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
26	25	neneqd 2933	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 = 2)
27		prmuz2 16607	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
28	7, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
29		2prm 16603	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
30		dvdsprm 16614	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
31	28, 29, 30	sylancl 586	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
32	26, 31	mtbird 325	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 ∥ 2)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ 2)
34		1cnd 11107	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → 1 ∈ ℂ)
35	2	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℕ)
36		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2))
37		oexpneg 16256	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℂ ∧ ((𝑃 − 1) / 2) ∈ ℕ ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
38	34, 35, 36, 37	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
39	35	nnzd 12495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℤ)
40		1exp 13998	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → (1↑((𝑃 − 1) / 2)) = 1)
41	39, 40	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (1↑((𝑃 − 1) / 2)) = 1)
42	41	negeqd 11354	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1)
43	38, 42	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1)
44	43	oveq1d 7361	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 + 1))
45		ax-1cn 11064	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46		neg1cn 12110	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
47		1pneg1e0 12239	. . . . . . . . . . . . . 14 ⊢ (1 + -1) = 0
48	45, 46, 47	addcomli 11305	. . . . . . . . . . . . 13 ⊢ (-1 + 1) = 0
49	44, 48	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0)
50	49	oveq2d 7362	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 − 0))
51		2cn 12200	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
52	51	subid1i 11433	. . . . . . . . . . 11 ⊢ (2 − 0) = 2
53	50, 52	eqtrdi 2782	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = 2)
54	53	breq2d 5101	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2))
55	33, 54	mtbird 325	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)))
56	55	ex 412	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (¬ 2 ∥ ((𝑃 − 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
57	56	con4d 115	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2)))
58		2z 12504	. . . . . . . 8 ⊢ 2 ∈ ℤ
59	58	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℤ)
60		moddvds 16174	. . . . . . 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))))
61	9, 59, 6, 60	syl3anc 1373	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
62		4z 12506	. . . . . . . . 9 ⊢ 4 ∈ ℤ
63		4ne0 12233	. . . . . . . . 9 ⊢ 4 ≠ 0
64		nnm1nn0 12422	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
65	9, 64	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℕ₀)
66	65	nn0zd 12494	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℤ)
67		dvdsval2 16166	. . . . . . . . 9 ⊢ ((4 ∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
68	62, 63, 66, 67	mp3an12i 1467	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
69	65	nn0cnd 12444	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℂ)
70	51	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℂ)
71		2ne0 12229	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ≠ 0)
73	69, 70, 70, 72, 72	divdiv1d 11928	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / (2 · 2)))
74		2t2e4 12284	. . . . . . . . . . 11 ⊢ (2 · 2) = 4
75	74	oveq2i 7357	. . . . . . . . . 10 ⊢ ((𝑃 − 1) / (2 · 2)) = ((𝑃 − 1) / 4)
76	73, 75	eqtrdi 2782	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / 4))
77	76	eleq1d 2816	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((((𝑃 − 1) / 2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈ ℤ))
78	68, 77	bitr4d 282	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
79	2	nnzd 12495	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ)
80		dvdsval2 16166	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
81	58, 71, 79, 80	mp3an12i 1467	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
82	78, 81	bitr4d 282	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ 2 ∥ ((𝑃 − 1) / 2)))
83	57, 61, 82	3imtr4d 294	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) → 4 ∥ (𝑃 − 1)))
84	46	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → -1 ∈ ℂ)
85		neg1ne0 12112	. . . . . . . . . . . 12 ⊢ -1 ≠ 0
86	85	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → -1 ≠ 0)
87	58	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 ∈ ℤ)
88	78	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (((𝑃 − 1) / 2) / 2) ∈ ℤ)
89		expmulz 14015	. . . . . . . . . . 11 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ (((𝑃 − 1) / 2) / 2) ∈ ℤ)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
90	84, 86, 87, 88, 89	syl22anc 838	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
91	2	nncnd 12141	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℂ)
92	91, 70, 72	divcan2d 11899	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
93	92	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
94	93	oveq2d 7362	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) / 2)))
95		neg1sqe1 14103	. . . . . . . . . . . 12 ⊢ (-1↑2) = 1
96	95	oveq1i 7356	. . . . . . . . . . 11 ⊢ ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) / 2))
97		1exp 13998	. . . . . . . . . . . 12 ⊢ ((((𝑃 − 1) / 2) / 2) ∈ ℤ → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
98	88, 97	syl 17	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
99	96, 98	eqtrid 2778	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1)
100	90, 94, 99	3eqtr3d 2774	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑((𝑃 − 1) / 2)) = 1)
101	100	oveq1d 7361	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (1 + 1))
102	22, 101	eqtr4id 2785	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1))
103	102	oveq1d 7361	. . . . . 6 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃))
104	103	ex 412	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
105	83, 104	impbid 212	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 4 ∥ (𝑃 − 1)))
106	13, 24, 105	3bitr2d 307	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1)))
107		lgsval3 27253	. . . . 5 ⊢ ((-1 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
108	1, 107	mpan 690	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
109	108	eqeq1d 2733	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1))
110		4nn 12208	. . . . 5 ⊢ 4 ∈ ℕ
111	110	a1i 11	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 4 ∈ ℕ)
112		prmz 16586	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
113	7, 112	syl 17	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
114		1zzd 12503	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℤ)
115		moddvds 16174	. . . 4 ⊢ ((4 ∈ ℕ ∧ 𝑃 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
116	111, 113, 114, 115	syl3anc 1373	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
117	106, 109, 116	3bitr4d 311	. 2 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4)))
118		1re 11112	. . . 4 ⊢ 1 ∈ ℝ
119		nnrp 12902	. . . . 5 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
120	110, 119	ax-mp 5	. . . 4 ⊢ 4 ∈ ℝ⁺
121		0le1 11640	. . . 4 ⊢ 0 ≤ 1
122		1lt4 12296	. . . 4 ⊢ 1 < 4
123		modid 13800	. . . 4 ⊢ (((1 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 1 ∧ 1 < 4)) → (1 mod 4) = 1)
124	118, 120, 121, 122, 123	mp4an 693	. . 3 ⊢ (1 mod 4) = 1
125	124	eqeq2i 2744	. 2 ⊢ ((𝑃 mod 4) = (1 mod 4) ↔ (𝑃 mod 4) = 1)
126	117, 125	bitrdi 287	1 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∖ cdif 3894 {csn 4573 class class class wbr 5089 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 < clt 11146 ≤ cle 11147 − cmin 11344 -cneg 11345 / cdiv 11774 ℕcn 12125 2c2 12180 4c4 12182 ℕ₀cn0 12381 ℤcz 12468 ℤ_≥cuz 12732 ℝ⁺crp 12890 mod cmo 13773 ↑cexp 13968 ∥ cdvds 16163 ℙcprime 16582 /_L clgs 27232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-n0 12382 df-xnn0 12455 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-dvds 16164 df-gcd 16406 df-prm 16583 df-phi 16677 df-pc 16749 df-lgs 27233

This theorem is referenced by: 2sqlem11 27367 2sqblem 27369

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