2lgs - Intuitionistic Logic Explorer

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Theorem 2lgs 15748

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 15661) 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 12614	. 2 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
2		2lgslem4 15747	. . . . . 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 5982	. . . . . 6 ⊢ (𝑃 = 2 → (2 /_L 𝑃) = (2 /_L 2))
5	4	eqeq1d 2218	. . . . 5 ⊢ (𝑃 = 2 → ((2 /_L 𝑃) = 1 ↔ (2 /_L 2) = 1))
6		oveq1 5981	. . . . . 6 ⊢ (𝑃 = 2 → (𝑃 mod 8) = (2 mod 8))
7	6	eleq1d 2278	. . . . 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 12615	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
11		prmnn 12598	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		dvdsprime 12610	. . . . . . . . . 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 12391	. . . . . . . . . . . . 13 ⊢ 2 ∥ 2
15		breq2 4066	. . . . . . . . . . . . 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 2272	. . . . . . . . . . . 12 ⊢ (𝑃 = 1 → (𝑃 ∈ ℙ ↔ 1 ∈ ℙ))
19		1nprm 12602	. . . . . . . . . . . . 13 ⊢ ¬ 1 ∈ ℙ
20	19	pm2.21i 649	. . . . . . . . . . . 12 ⊢ (1 ∈ ℙ → 2 ∥ 𝑃)
21	18, 20	biimtrdi 163	. . . . . . . . . . 11 ⊢ (𝑃 = 1 → (𝑃 ∈ ℙ → 2 ∥ 𝑃))
22	17, 21	jaoi 720	. . . . . . . . . 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 638	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ¬ 𝑃 ∥ 2)
26		2z 9442	. . . . . . 7 ⊢ 2 ∈ ℤ
27	25, 26	jctil 312	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2))
28		2lgslem1 15735	. . . . . . 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 2215	. . . . . 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 12751	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃))
31	30	biimpri 133	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2}))
32	31	3ad2ant1 1023	. . . . . . . 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 2209	. . . . . . . 8 ⊢ ((𝑃 − 1) / 2) = ((𝑃 − 1) / 2)
34		eqid 2209	. . . . . . . 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 2209	. . . . . . . 8 ⊢ (⌊‘(𝑃 / 4)) = (⌊‘(𝑃 / 4))
36		eqid 2209	. . . . . . . 8 ⊢ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))
37	32, 33, 34, 35, 36	gausslemma2d 15713	. . . . . . 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 2218	. . . . . 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 1354	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((2 /_L 𝑃) = 1 ↔ (-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1))
40	36	2lgslem2 15736	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ)
41		m1exp1 12378	. . . . . 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 9240	. . . . . . 7 ⊢ 2 ∈ ℕ
44		dvdsval3 12268	. . . . . . 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 15745	. . . . . . . 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 2218	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = 0 ↔ if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0))
49		prmz 12599	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
50		8nn 9246	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ
51	50	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 8 ∈ ℕ)
52	49, 51	zmodcld 10534	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℕ₀)
53	52	nn0zd 9535	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℤ)
54		1z 9440	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
55		zdceq 9490	. . . . . . . . . . . . 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 9245	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℕ
58	57	nnzi 9435	. . . . . . . . . . . . 13 ⊢ 7 ∈ ℤ
59		zdceq 9490	. . . . . . . . . . . . 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 940	. . . . . . . . . . . 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 3666	. . . . . . . . . . . . 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 842	. . . . . . . . . . 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 840	. . . . . . . . . 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 3590	. . . . . . . . . . . 12 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
70	69	eqeq1d 2218	. . . . . . . . . . 11 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 ↔ 1 = 0))
71		1ne0 9146	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
72		eqneqall 2390	. . . . . . . . . . . 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 800	. . . . . . . . 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 3587	. . . . . . . 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 720	. 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 712 DECID wdc 838 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ≠ wne 2380 ∃wrex 2489 {crab 2492 ∖ cdif 3174 ifcif 3582 {csn 3646 {cpr 3647 class class class wbr 4062 ↦ cmpt 4124 ‘cfv 5294 (class class class)co 5974 0cc0 7967 1c1 7968 · cmul 7972 < clt 8149 − cmin 8285 -cneg 8286 / cdiv 8787 ℕcn 9078 2c2 9129 4c4 9131 7c7 9134 8c8 9135 ℕ₀cn0 9337 ℤcz 9414 ...cfz 10172 ⌊cfl 10455 mod cmo 10511 ↑cexp 10727 ♯chash 10964 ∥ cdvds 12264 ℙcprime 12595 /_L clgs 15641

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-stab 835 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-xor 1398 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-tp 3654 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-irdg 6486 df-frec 6507 df-1o 6532 df-2o 6533 df-oadd 6536 df-er 6650 df-en 6858 df-dom 6859 df-fin 6860 df-sup 7119 df-inf 7120 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-5 9140 df-6 9141 df-7 9142 df-8 9143 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-ioo 10056 df-ico 10058 df-fz 10173 df-fzo 10307 df-fl 10457 df-mod 10512 df-seqfrec 10637 df-exp 10728 df-fac 10915 df-ihash 10965 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-clim 11756 df-proddc 12028 df-dvds 12265 df-gcd 12441 df-prm 12596 df-phi 12699 df-pc 12774 df-lgs 15642

This theorem is referenced by: 2lgsoddprm 15757

W3C validator