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Theorem m1lgs 27518

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 12630	. . . . . . . . 9 ⊢ -1 ∈ ℤ
2		oddprm 16870	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	nnnn0d 12565	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ₀)
4		zexpcl 14112	. . . . . . . . 9 ⊢ ((-1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
5	1, 3, 4	sylancr 598	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
6	5	peano2zd 12703	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
7		eldifi 4093	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16732	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
10	6, 9	zmodcld 13925	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
11	10	nn0cnd 12567	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℂ)
12		1cnd 11202	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℂ)
13	11, 12, 12	subaddd 11587	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
14		2re 12315	. . . . . . . 8 ⊢ 2 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℝ)
16	9	nnrpd 13058	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ⁺)
17		0le2 12343	. . . . . . . 8 ⊢ 0 ≤ 2
18	17	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 0 ≤ 2)
19		oddprmgt2 16758	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
20		modid 13929	. . . . . . 7 ⊢ (((2 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
21	15, 16, 18, 19, 20	syl22anc 851	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = 2)
22		df-2 12303	. . . . . 6 ⊢ 2 = (1 + 1)
23	21, 22	eqtrdi 2820	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = (1 + 1))
24	23	eqeq1d 2771	. . . 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 4762	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
26	25	neneqd 2969	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 = 2)
27		prmuz2 16754	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
28	7, 27	syl 18	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
29		2prm 16750	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
30		dvdsprm 16762	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
31	28, 29, 30	sylancl 597	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
32	26, 31	mtbird 328	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 ∥ 2)
33	32	adantr 485	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ 2)
34		1cnd 11202	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → 1 ∈ ℂ)
35	2	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℕ)
36		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2))
37		oexpneg 16403	. . . . . . . . . . . . . . . 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 1396	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
39	35	nnzd 12617	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℤ)
40		1exp 14127	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 − 1) / 2) ∈ ℤ → (1↑((𝑃 − 1) / 2)) = 1)
41	39, 40	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (1↑((𝑃 − 1) / 2)) = 1)
42	41	negeqd 11451	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1)
43	38, 42	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1)
44	43	oveq1d 7426	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 + 1))
45		ax-1cn 11158	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46		neg1cn 12203	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
47		1pneg1e0 12358	. . . . . . . . . . . . . 14 ⊢ (1 + -1) = 0
48	45, 46, 47	addcomli 11402	. . . . . . . . . . . . 13 ⊢ (-1 + 1) = 0
49	44, 48	eqtrdi 2820	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0)
50	49	oveq2d 7427	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 − 0))
51		2cn 12316	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
52	51	subid1i 11530	. . . . . . . . . . 11 ⊢ (2 − 0) = 2
53	50, 52	eqtrdi 2820	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = 2)
54	53	breq2d 5125	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2))
55	33, 54	mtbird 328	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)))
56	55	ex 417	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (¬ 2 ∥ ((𝑃 − 1) / 2) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
57	56	con4d 116	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) → 2 ∥ ((𝑃 − 1) / 2)))
58		2z 12626	. . . . . . . 8 ⊢ 2 ∈ ℤ
59	58	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℤ)
60		moddvds 16321	. . . . . . 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 1396	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
62		4z 12628	. . . . . . . . 9 ⊢ 4 ∈ ℤ
63		4ne0 12352	. . . . . . . . 9 ⊢ 4 ≠ 0
64		nnm1nn0 12545	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
65	9, 64	syl 18	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℕ₀)
66	65	nn0zd 12616	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℤ)
67		dvdsval2 16313	. . . . . . . . 9 ⊢ ((4 ∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
68	62, 63, 66, 67	mp3an12i 1491	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
69	65	nn0cnd 12567	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℂ)
70	51	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℂ)
71		2ne0 12347	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ≠ 0)
73	69, 70, 70, 72, 72	divdiv1d 12022	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / (2 · 2)))
74		2t2e4 12404	. . . . . . . . . . 11 ⊢ (2 · 2) = 4
75	74	oveq2i 7422	. . . . . . . . . 10 ⊢ ((𝑃 − 1) / (2 · 2)) = ((𝑃 − 1) / 4)
76	73, 75	eqtrdi 2820	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / 4))
77	76	eleq1d 2854	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((((𝑃 − 1) / 2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈ ℤ))
78	68, 77	bitr4d 285	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
79	2	nnzd 12617	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ)
80		dvdsval2 16313	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
81	58, 71, 79, 80	mp3an12i 1491	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
82	78, 81	bitr4d 285	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ 2 ∥ ((𝑃 − 1) / 2)))
83	57, 61, 82	3imtr4d 297	. . . . 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 12205	. . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (((𝑃 − 1) / 2) / 2) ∈ ℤ)
89		expmulz 14144	. . . . . . . . . . 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 851	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
91	2	nncnd 12249	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℂ)
92	91, 70, 72	divcan2d 11993	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
93	92	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
94	93	oveq2d 7427	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) / 2)))
95		neg1sqe1 14232	. . . . . . . . . . . 12 ⊢ (-1↑2) = 1
96	95	oveq1i 7421	. . . . . . . . . . 11 ⊢ ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) / 2))
97		1exp 14127	. . . . . . . . . . . 12 ⊢ ((((𝑃 − 1) / 2) / 2) ∈ ℤ → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
98	88, 97	syl 18	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (1↑(((𝑃 − 1) / 2) / 2)) = 1)
99	96, 98	eqtrid 2816	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1)
100	90, 94, 99	3eqtr3d 2812	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑((𝑃 − 1) / 2)) = 1)
101	100	oveq1d 7426	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (1 + 1))
102	22, 101	eqtr4id 2823	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1))
103	102	oveq1d 7426	. . . . . 6 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃))
104	103	ex 417	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
105	83, 104	impbid 215	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 4 ∥ (𝑃 − 1)))
106	13, 24, 105	3bitr2d 310	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1)))
107		lgsval3 27445	. . . . 5 ⊢ ((-1 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
108	1, 107	mpan 702	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
109	108	eqeq1d 2771	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1))
110		4nn 12324	. . . . 5 ⊢ 4 ∈ ℕ
111	110	a1i 11	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 4 ∈ ℕ)
112		prmz 16733	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
113	7, 112	syl 18	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
114		1zzd 12625	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℤ)
115		moddvds 16321	. . . 4 ⊢ ((4 ∈ ℕ ∧ 𝑃 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
116	111, 113, 114, 115	syl3anc 1396	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
117	106, 109, 116	3bitr4d 314	. 2 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4)))
118		1re 11208	. . . 4 ⊢ 1 ∈ ℝ
119		nnrp 13028	. . . . 5 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
120	110, 119	ax-mp 5	. . . 4 ⊢ 4 ∈ ℝ⁺
121		0le1 11737	. . . 4 ⊢ 0 ≤ 1
122		1lt4 12419	. . . 4 ⊢ 1 < 4
123		modid 13929	. . . 4 ⊢ (((1 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 1 ∧ 1 < 4)) → (1 mod 4) = 1)
124	118, 120, 121, 122, 123	mp4an 705	. . 3 ⊢ (1 mod 4) = 1
125	124	eqeq2i 2782	. 2 ⊢ ((𝑃 mod 4) = (1 mod 4) ↔ (𝑃 mod 4) = 1)
126	117, 125	bitrdi 290	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 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3910 {csn 4594 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 ≤ cle 11244 − cmin 11441 -cneg 11442 / cdiv 11871 ℕcn 12233 2c2 12295 4c4 12297 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 ℝ⁺crp 13016 mod cmo 13902 ↑cexp 14097 ∥ cdvds 16310 ℙcprime 16729 /_L clgs 27424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-inf 9403 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-n0 12505 df-xnn0 12578 df-z 12592 df-uz 12863 df-q 12973 df-rp 13017 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553 df-prm 16730 df-phi 16825 df-pc 16897 df-lgs 27425

This theorem is referenced by: 2sqlem11 27559 2sqblem 27561

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