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Theorem m1lgs 27341

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 12636	. . . . . . . . 9 ⊢ -1 ∈ ℤ
2		oddprm 16786	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
3	2	nnnn0d 12570	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ₀)
4		zexpcl 14081	. . . . . . . . 9 ⊢ ((-1 ∈ ℤ ∧ ((𝑃 − 1) / 2) ∈ ℕ₀) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
5	1, 3, 4	sylancr 585	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1↑((𝑃 − 1) / 2)) ∈ ℤ)
6	5	peano2zd 12707	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1↑((𝑃 − 1) / 2)) + 1) ∈ ℤ)
7		eldifi 4127	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
8		prmnn 16652	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℕ)
10	6, 9	zmodcld 13897	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℕ₀)
11	10	nn0cnd 12572	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ ℂ)
12		1cnd 11247	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℂ)
13	11, 12, 12	subaddd 11627	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ (1 + 1) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
14		2re 12324	. . . . . . . 8 ⊢ 2 ∈ ℝ
15	14	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℝ)
16	9	nnrpd 13054	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℝ⁺)
17		0le2 12352	. . . . . . . 8 ⊢ 0 ≤ 2
18	17	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 0 ≤ 2)
19		oddprmgt2 16677	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 < 𝑃)
20		modid 13901	. . . . . . 7 ⊢ (((2 ∈ ℝ ∧ 𝑃 ∈ ℝ⁺) ∧ (0 ≤ 2 ∧ 2 < 𝑃)) → (2 mod 𝑃) = 2)
21	15, 16, 18, 19, 20	syl22anc 837	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = 2)
22		df-2 12313	. . . . . 6 ⊢ 2 = (1 + 1)
23	21, 22	eqtrdi 2784	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 mod 𝑃) = (1 + 1))
24	23	eqeq1d 2730	. . . 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 4798	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ≠ 2)
26	25	neneqd 2942	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 = 2)
27		prmuz2 16674	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘2))
28	7, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ (ℤ_≥‘2))
29		2prm 16670	. . . . . . . . . . . 12 ⊢ 2 ∈ ℙ
30		dvdsprm 16681	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℤ_≥‘2) ∧ 2 ∈ ℙ) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
31	28, 29, 30	sylancl 584	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 ∥ 2 ↔ 𝑃 = 2))
32	26, 31	mtbird 324	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 𝑃 ∥ 2)
33	32	adantr 479	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ 2)
34		1cnd 11247	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → 1 ∈ ℂ)
35	2	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℕ)
36		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 2 ∥ ((𝑃 − 1) / 2))
37		oexpneg 16329	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -(1↑((𝑃 − 1) / 2)))
39	35	nnzd 12623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((𝑃 − 1) / 2) ∈ ℤ)
40		1exp 14096	. . . . . . . . . . . . . . . . 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 11492	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → -(1↑((𝑃 − 1) / 2)) = -1)
43	38, 42	eqtrd 2768	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (-1↑((𝑃 − 1) / 2)) = -1)
44	43	oveq1d 7441	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (-1 + 1))
45		ax-1cn 11204	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
46		neg1cn 12364	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℂ
47		1pneg1e0 12369	. . . . . . . . . . . . . 14 ⊢ (1 + -1) = 0
48	45, 46, 47	addcomli 11444	. . . . . . . . . . . . 13 ⊢ (-1 + 1) = 0
49	44, 48	eqtrdi 2784	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ((-1↑((𝑃 − 1) / 2)) + 1) = 0)
50	49	oveq2d 7442	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = (2 − 0))
51		2cn 12325	. . . . . . . . . . . 12 ⊢ 2 ∈ ℂ
52	51	subid1i 11570	. . . . . . . . . . 11 ⊢ (2 − 0) = 2
53	50, 52	eqtrdi 2784	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) = 2)
54	53	breq2d 5164	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → (𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)) ↔ 𝑃 ∥ 2))
55	33, 54	mtbird 324	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 2 ∥ ((𝑃 − 1) / 2)) → ¬ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1)))
56	55	ex 411	. . . . . . 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 12632	. . . . . . . 8 ⊢ 2 ∈ ℤ
59	58	a1i 11	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℤ)
60		moddvds 16249	. . . . . . 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 1368	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 𝑃 ∥ (2 − ((-1↑((𝑃 − 1) / 2)) + 1))))
62		4z 12634	. . . . . . . . 9 ⊢ 4 ∈ ℤ
63		4ne0 12358	. . . . . . . . 9 ⊢ 4 ≠ 0
64		nnm1nn0 12551	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℕ → (𝑃 − 1) ∈ ℕ₀)
65	9, 64	syl 17	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℕ₀)
66	65	nn0zd 12622	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℤ)
67		dvdsval2 16241	. . . . . . . . 9 ⊢ ((4 ∈ ℤ ∧ 4 ≠ 0 ∧ (𝑃 − 1) ∈ ℤ) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
68	62, 63, 66, 67	mp3an12i 1461	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ ((𝑃 − 1) / 4) ∈ ℤ))
69	65	nn0cnd 12572	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (𝑃 − 1) ∈ ℂ)
70	51	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ∈ ℂ)
71		2ne0 12354	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
72	71	a1i 11	. . . . . . . . . . 11 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 2 ≠ 0)
73	69, 70, 70, 72, 72	divdiv1d 12059	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / (2 · 2)))
74		2t2e4 12414	. . . . . . . . . . 11 ⊢ (2 · 2) = 4
75	74	oveq2i 7437	. . . . . . . . . 10 ⊢ ((𝑃 − 1) / (2 · 2)) = ((𝑃 − 1) / 4)
76	73, 75	eqtrdi 2784	. . . . . . . . 9 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((𝑃 − 1) / 2) / 2) = ((𝑃 − 1) / 4))
77	76	eleq1d 2814	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((((𝑃 − 1) / 2) / 2) ∈ ℤ ↔ ((𝑃 − 1) / 4) ∈ ℤ))
78	68, 77	bitr4d 281	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
79	2	nnzd 12623	. . . . . . . 8 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℤ)
80		dvdsval2 16241	. . . . . . . 8 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ ((𝑃 − 1) / 2) ∈ ℤ) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
81	58, 71, 79, 80	mp3an12i 1461	. . . . . . 7 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 ∥ ((𝑃 − 1) / 2) ↔ (((𝑃 − 1) / 2) / 2) ∈ ℤ))
82	78, 81	bitr4d 281	. . . . . 6 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) ↔ 2 ∥ ((𝑃 − 1) / 2)))
83	57, 61, 82	3imtr4d 293	. . . . 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 12366	. . . . . . . . . . . 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 475	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (((𝑃 − 1) / 2) / 2) ∈ ℤ)
89		expmulz 14113	. . . . . . . . . . 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 837	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = ((-1↑2)↑(((𝑃 − 1) / 2) / 2)))
91	2	nncnd 12266	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℂ)
92	91, 70, 72	divcan2d 12030	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
93	92	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 · (((𝑃 − 1) / 2) / 2)) = ((𝑃 − 1) / 2))
94	93	oveq2d 7442	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑(2 · (((𝑃 − 1) / 2) / 2))) = (-1↑((𝑃 − 1) / 2)))
95		neg1sqe1 14199	. . . . . . . . . . . 12 ⊢ (-1↑2) = 1
96	95	oveq1i 7436	. . . . . . . . . . 11 ⊢ ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = (1↑(((𝑃 − 1) / 2) / 2))
97		1exp 14096	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑2)↑(((𝑃 − 1) / 2) / 2)) = 1)
100	90, 94, 99	3eqtr3d 2776	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (-1↑((𝑃 − 1) / 2)) = 1)
101	100	oveq1d 7441	. . . . . . . 8 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → ((-1↑((𝑃 − 1) / 2)) + 1) = (1 + 1))
102	22, 101	eqtr4id 2787	. . . . . . 7 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → 2 = ((-1↑((𝑃 − 1) / 2)) + 1))
103	102	oveq1d 7441	. . . . . 6 ⊢ ((𝑃 ∈ (ℙ ∖ {2}) ∧ 4 ∥ (𝑃 − 1)) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃))
104	103	ex 411	. . . . 5 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (4 ∥ (𝑃 − 1) → (2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃)))
105	83, 104	impbid 211	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((2 mod 𝑃) = (((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ↔ 4 ∥ (𝑃 − 1)))
106	13, 24, 105	3bitr2d 306	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1 ↔ 4 ∥ (𝑃 − 1)))
107		lgsval3 27268	. . . . 5 ⊢ ((-1 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
108	1, 107	mpan 688	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → (-1 /_L 𝑃) = ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))
109	108	eqeq1d 2730	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ ((((-1↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) = 1))
110		4nn 12333	. . . . 5 ⊢ 4 ∈ ℕ
111	110	a1i 11	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 4 ∈ ℕ)
112		prmz 16653	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
113	7, 112	syl 17	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
114		1zzd 12631	. . . 4 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 1 ∈ ℤ)
115		moddvds 16249	. . . 4 ⊢ ((4 ∈ ℕ ∧ 𝑃 ∈ ℤ ∧ 1 ∈ ℤ) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
116	111, 113, 114, 115	syl3anc 1368	. . 3 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 mod 4) = (1 mod 4) ↔ 4 ∥ (𝑃 − 1)))
117	106, 109, 116	3bitr4d 310	. 2 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((-1 /_L 𝑃) = 1 ↔ (𝑃 mod 4) = (1 mod 4)))
118		1re 11252	. . . 4 ⊢ 1 ∈ ℝ
119		nnrp 13025	. . . . 5 ⊢ (4 ∈ ℕ → 4 ∈ ℝ⁺)
120	110, 119	ax-mp 5	. . . 4 ⊢ 4 ∈ ℝ⁺
121		0le1 11775	. . . 4 ⊢ 0 ≤ 1
122		1lt4 12426	. . . 4 ⊢ 1 < 4
123		modid 13901	. . . 4 ⊢ (((1 ∈ ℝ ∧ 4 ∈ ℝ⁺) ∧ (0 ≤ 1 ∧ 1 < 4)) → (1 mod 4) = 1)
124	118, 120, 121, 122, 123	mp4an 691	. . 3 ⊢ (1 mod 4) = 1
125	124	eqeq2i 2741	. 2 ⊢ ((𝑃 mod 4) = (1 mod 4) ↔ (𝑃 mod 4) = 1)
126	117, 125	bitrdi 286	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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∖ cdif 3946 {csn 4632 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 < clt 11286 ≤ cle 11287 − cmin 11482 -cneg 11483 / cdiv 11909 ℕcn 12250 2c2 12305 4c4 12307 ℕ₀cn0 12510 ℤcz 12596 ℤ_≥cuz 12860 ℝ⁺crp 13014 mod cmo 13874 ↑cexp 14066 ∥ cdvds 16238 ℙcprime 16649 /_L clgs 27247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-dvds 16239 df-gcd 16477 df-prm 16650 df-phi 16742 df-pc 16813 df-lgs 27248

This theorem is referenced by: 2sqlem11 27382 2sqblem 27384

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