2lgs - Intuitionistic Logic Explorer

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Theorem 2lgs 15906

Description: The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime 𝑃 iff 𝑃≡±1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of (𝑁 /_L 2) (lgs2 15819) to some degree, by demanding that reciprocity extend to the case 𝑄 = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.)

**Assertion**
Ref	Expression
2lgs	⊢ (𝑃 ∈ ℙ → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7}))

Proof of Theorem 2lgs Dummy variables 𝑖 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prm2orodd 12761	. 2 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
2		2lgslem4 15905	. . . . . 6 ⊢ ((2 /_L 2) = 1 ↔ (2 mod 8) ∈ {1, 7})
3	2	a1i 9	. . . . 5 ⊢ (𝑃 = 2 → ((2 /_L 2) = 1 ↔ (2 mod 8) ∈ {1, 7}))
4		oveq2 6036	. . . . . 6 ⊢ (𝑃 = 2 → (2 /_L 𝑃) = (2 /_L 2))
5	4	eqeq1d 2240	. . . . 5 ⊢ (𝑃 = 2 → ((2 /_L 𝑃) = 1 ↔ (2 /_L 2) = 1))
6		oveq1 6035	. . . . . 6 ⊢ (𝑃 = 2 → (𝑃 mod 8) = (2 mod 8))
7	6	eleq1d 2300	. . . . 5 ⊢ (𝑃 = 2 → ((𝑃 mod 8) ∈ {1, 7} ↔ (2 mod 8) ∈ {1, 7}))
8	3, 5, 7	3bitr4d 220	. . . 4 ⊢ (𝑃 = 2 → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7}))
9	8	a1d 22	. . 3 ⊢ (𝑃 = 2 → (𝑃 ∈ ℙ → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7})))
10		2prm 12762	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
11		prmnn 12745	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		dvdsprime 12757	. . . . . . . . . 10 ⊢ ((2 ∈ ℙ ∧ 𝑃 ∈ ℕ) → (𝑃 ∥ 2 ↔ (𝑃 = 2 ∨ 𝑃 = 1)))
13	10, 11, 12	sylancr 414	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → (𝑃 ∥ 2 ↔ (𝑃 = 2 ∨ 𝑃 = 1)))
14		z2even 12538	. . . . . . . . . . . . 13 ⊢ 2 ∥ 2
15		breq2 4097	. . . . . . . . . . . . 13 ⊢ (𝑃 = 2 → (2 ∥ 𝑃 ↔ 2 ∥ 2))
16	14, 15	mpbiri 168	. . . . . . . . . . . 12 ⊢ (𝑃 = 2 → 2 ∥ 𝑃)
17	16	a1d 22	. . . . . . . . . . 11 ⊢ (𝑃 = 2 → (𝑃 ∈ ℙ → 2 ∥ 𝑃))
18		eleq1 2294	. . . . . . . . . . . 12 ⊢ (𝑃 = 1 → (𝑃 ∈ ℙ ↔ 1 ∈ ℙ))
19		1nprm 12749	. . . . . . . . . . . . 13 ⊢ ¬ 1 ∈ ℙ
20	19	pm2.21i 651	. . . . . . . . . . . 12 ⊢ (1 ∈ ℙ → 2 ∥ 𝑃)
21	18, 20	biimtrdi 163	. . . . . . . . . . 11 ⊢ (𝑃 = 1 → (𝑃 ∈ ℙ → 2 ∥ 𝑃))
22	17, 21	jaoi 724	. . . . . . . . . 10 ⊢ ((𝑃 = 2 ∨ 𝑃 = 1) → (𝑃 ∈ ℙ → 2 ∥ 𝑃))
23	22	com12 30	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → ((𝑃 = 2 ∨ 𝑃 = 1) → 2 ∥ 𝑃))
24	13, 23	sylbid 150	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (𝑃 ∥ 2 → 2 ∥ 𝑃))
25	24	con3dimp 640	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ¬ 𝑃 ∥ 2)
26		2z 9552	. . . . . . 7 ⊢ 2 ∈ ℤ
27	25, 26	jctil 312	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2))
28		2lgslem1 15893	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))))
29	28	eqcomd 2237	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))}))
30		nnoddn2prmb 12898	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃))
31	30	biimpri 133	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2}))
32	31	3ad2ant1 1045	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) ∧ (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2) ∧ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))})) → 𝑃 ∈ (ℙ ∖ {2}))
33		eqid 2231	. . . . . . . 8 ⊢ ((𝑃 − 1) / 2) = ((𝑃 − 1) / 2)
34		eqid 2231	. . . . . . . 8 ⊢ (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2)))) = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
35		eqid 2231	. . . . . . . 8 ⊢ (⌊‘(𝑃 / 4)) = (⌊‘(𝑃 / 4))
36		eqid 2231	. . . . . . . 8 ⊢ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))
37	32, 33, 34, 35, 36	gausslemma2d 15871	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) ∧ (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2) ∧ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))})) → (2 /_L 𝑃) = (-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))))
38	37	eqeq1d 2240	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) ∧ (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2) ∧ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (♯‘{𝑥 ∈ ℤ ∣ ∃𝑖 ∈ (1...((𝑃 − 1) / 2))(𝑥 = (𝑖 · 2) ∧ (𝑃 / 2) < (𝑥 mod 𝑃))})) → ((2 /_L 𝑃) = 1 ↔ (-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1))
39	27, 29, 38	mpd3an23 1376	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((2 /_L 𝑃) = 1 ↔ (-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1))
40	36	2lgslem2 15894	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ)
41		m1exp1 12525	. . . . . 6 ⊢ ((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ → ((-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1 ↔ 2 ∥ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))))
42	40, 41	syl 14	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1 ↔ 2 ∥ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))))
43		2nn 9348	. . . . . . 7 ⊢ 2 ∈ ℕ
44		dvdsval3 12415	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ) → (2 ∥ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ↔ ((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = 0))
45	43, 40, 44	sylancr 414	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∥ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ↔ ((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = 0))
46	36	2lgslem3 15903	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ ∧ ¬ 2 ∥ 𝑃) → ((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))
47	11, 46	sylan 283	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))
48	47	eqeq1d 2240	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = 0 ↔ if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0))
49		prmz 12746	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
50		8nn 9354	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ
51	50	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 8 ∈ ℕ)
52	49, 51	zmodcld 10653	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℕ₀)
53	52	nn0zd 9645	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℤ)
54		1z 9550	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
55		zdceq 9600	. . . . . . . . . . . . 13 ⊢ (((𝑃 mod 8) ∈ ℤ ∧ 1 ∈ ℤ) → DECID (𝑃 mod 8) = 1)
56	53, 54, 55	sylancl 413	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → DECID (𝑃 mod 8) = 1)
57		7nn 9353	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℕ
58	57	nnzi 9545	. . . . . . . . . . . . 13 ⊢ 7 ∈ ℤ
59		zdceq 9600	. . . . . . . . . . . . 13 ⊢ (((𝑃 mod 8) ∈ ℤ ∧ 7 ∈ ℤ) → DECID (𝑃 mod 8) = 7)
60	53, 58, 59	sylancl 413	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → DECID (𝑃 mod 8) = 7)
61		dcor 944	. . . . . . . . . . . 12 ⊢ (DECID (𝑃 mod 8) = 1 → (DECID (𝑃 mod 8) = 7 → DECID ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7)))
62	56, 60, 61	sylc 62	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → DECID ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7))
63		elprg 3693	. . . . . . . . . . . . 13 ⊢ ((𝑃 mod 8) ∈ ℕ₀ → ((𝑃 mod 8) ∈ {1, 7} ↔ ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7)))
64	52, 63	syl 14	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 8) ∈ {1, 7} ↔ ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7)))
65	64	dcbid 846	. . . . . . . . . . 11 ⊢ (𝑃 ∈ ℙ → (DECID (𝑃 mod 8) ∈ {1, 7} ↔ DECID ((𝑃 mod 8) = 1 ∨ (𝑃 mod 8) = 7)))
66	62, 65	mpbird 167	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → DECID (𝑃 mod 8) ∈ {1, 7})
67		exmiddc 844	. . . . . . . . . 10 ⊢ (DECID (𝑃 mod 8) ∈ {1, 7} → ((𝑃 mod 8) ∈ {1, 7} ∨ ¬ (𝑃 mod 8) ∈ {1, 7}))
68	66, 67	syl 14	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → ((𝑃 mod 8) ∈ {1, 7} ∨ ¬ (𝑃 mod 8) ∈ {1, 7}))
69		iffalse 3617	. . . . . . . . . . . 12 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
70	69	eqeq1d 2240	. . . . . . . . . . 11 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 ↔ 1 = 0))
71		1ne0 9254	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
72		eqneqall 2413	. . . . . . . . . . . 12 ⊢ (1 = 0 → (1 ≠ 0 → (𝑃 mod 8) ∈ {1, 7}))
73	71, 72	mpi 15	. . . . . . . . . . 11 ⊢ (1 = 0 → (𝑃 mod 8) ∈ {1, 7})
74	70, 73	biimtrdi 163	. . . . . . . . . 10 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 → (𝑃 mod 8) ∈ {1, 7}))
75	74	jao1i 804	. . . . . . . . 9 ⊢ (((𝑃 mod 8) ∈ {1, 7} ∨ ¬ (𝑃 mod 8) ∈ {1, 7}) → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 → (𝑃 mod 8) ∈ {1, 7}))
76	68, 75	syl 14	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 → (𝑃 mod 8) ∈ {1, 7}))
77		iftrue 3614	. . . . . . . 8 ⊢ ((𝑃 mod 8) ∈ {1, 7} → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0)
78	76, 77	impbid1 142	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 ↔ (𝑃 mod 8) ∈ {1, 7}))
79	78	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 ↔ (𝑃 mod 8) ∈ {1, 7}))
80	45, 48, 79	3bitrd 214	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∥ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ↔ (𝑃 mod 8) ∈ {1, 7}))
81	39, 42, 80	3bitrd 214	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7}))
82	81	expcom 116	. . 3 ⊢ (¬ 2 ∥ 𝑃 → (𝑃 ∈ ℙ → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7})))
83	9, 82	jaoi 724	. 2 ⊢ ((𝑃 = 2 ∨ ¬ 2 ∥ 𝑃) → (𝑃 ∈ ℙ → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7})))
84	1, 83	mpcom 36	1 ⊢ (𝑃 ∈ ℙ → ((2 /_L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7}))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 ∃wrex 2512 {crab 2515 ∖ cdif 3198 ifcif 3607 {csn 3673 {cpr 3674 class class class wbr 4093 ↦ cmpt 4155 ‘cfv 5333 (class class class)co 6028 0cc0 8075 1c1 8076 · cmul 8080 < clt 8257 − cmin 8393 -cneg 8394 / cdiv 8895 ℕcn 9186 2c2 9237 4c4 9239 7c7 9242 8c8 9243 ℕ₀cn0 9445 ℤcz 9524 ...cfz 10288 ⌊cfl 10574 mod cmo 10630 ↑cexp 10846 ♯chash 11083 ∥ cdvds 12411 ℙcprime 12742 /_L clgs 15799

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-tp 3681 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-7 9250 df-8 9251 df-n0 9446 df-z 9525 df-uz 9801 df-q 9899 df-rp 9934 df-ioo 10172 df-ico 10174 df-fz 10289 df-fzo 10423 df-fl 10576 df-mod 10631 df-seqfrec 10756 df-exp 10847 df-fac 11034 df-ihash 11084 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-clim 11902 df-proddc 12175 df-dvds 12412 df-gcd 12588 df-prm 12743 df-phi 12846 df-pc 12921 df-lgs 15800

This theorem is referenced by: 2lgsoddprm 15915

W3C validator