2lgs - Intuitionistic Logic Explorer

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Theorem 2lgs 15804

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 15717) 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 12669	. 2 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
2		2lgslem4 15803	. . . . . 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 6018	. . . . . 6 ⊢ (𝑃 = 2 → (2 /_L 𝑃) = (2 /_L 2))
5	4	eqeq1d 2238	. . . . 5 ⊢ (𝑃 = 2 → ((2 /_L 𝑃) = 1 ↔ (2 /_L 2) = 1))
6		oveq1 6017	. . . . . 6 ⊢ (𝑃 = 2 → (𝑃 mod 8) = (2 mod 8))
7	6	eleq1d 2298	. . . . 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 12670	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
11		prmnn 12653	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		dvdsprime 12665	. . . . . . . . . 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 12446	. . . . . . . . . . . . 13 ⊢ 2 ∥ 2
15		breq2 4087	. . . . . . . . . . . . 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 2292	. . . . . . . . . . . 12 ⊢ (𝑃 = 1 → (𝑃 ∈ ℙ ↔ 1 ∈ ℙ))
19		1nprm 12657	. . . . . . . . . . . . 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 721	. . . . . . . . . 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 9490	. . . . . . 7 ⊢ 2 ∈ ℤ
27	25, 26	jctil 312	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2))
28		2lgslem1 15791	. . . . . . 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 2235	. . . . . 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 12806	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) ↔ (𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃))
31	30	biimpri 133	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → 𝑃 ∈ (ℙ ∖ {2}))
32	31	3ad2ant1 1042	. . . . . . . 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 2229	. . . . . . . 8 ⊢ ((𝑃 − 1) / 2) = ((𝑃 − 1) / 2)
34		eqid 2229	. . . . . . . 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 2229	. . . . . . . 8 ⊢ (⌊‘(𝑃 / 4)) = (⌊‘(𝑃 / 4))
36		eqid 2229	. . . . . . . 8 ⊢ (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) = (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))
37	32, 33, 34, 35, 36	gausslemma2d 15769	. . . . . . 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 2238	. . . . . 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 1373	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → ((2 /_L 𝑃) = 1 ↔ (-1↑(((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4)))) = 1))
40	36	2lgslem2 15792	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ)
41		m1exp1 12433	. . . . . 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 9288	. . . . . . 7 ⊢ 2 ∈ ℕ
44		dvdsval3 12323	. . . . . . 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 15801	. . . . . . . 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 2238	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) mod 2) = 0 ↔ if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0))
49		prmz 12654	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
50		8nn 9294	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ
51	50	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 8 ∈ ℕ)
52	49, 51	zmodcld 10584	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℕ₀)
53	52	nn0zd 9583	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℤ)
54		1z 9488	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
55		zdceq 9538	. . . . . . . . . . . . 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 9293	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℕ
58	57	nnzi 9483	. . . . . . . . . . . . 13 ⊢ 7 ∈ ℤ
59		zdceq 9538	. . . . . . . . . . . . 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 941	. . . . . . . . . . . 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 3686	. . . . . . . . . . . . 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 843	. . . . . . . . . . 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 841	. . . . . . . . . 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 3610	. . . . . . . . . . . 12 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 1)
70	69	eqeq1d 2238	. . . . . . . . . . 11 ⊢ (¬ (𝑃 mod 8) ∈ {1, 7} → (if((𝑃 mod 8) ∈ {1, 7}, 0, 1) = 0 ↔ 1 = 0))
71		1ne0 9194	. . . . . . . . . . . 12 ⊢ 1 ≠ 0
72		eqneqall 2410	. . . . . . . . . . . 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 801	. . . . . . . . 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 3607	. . . . . . . 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 721	. 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 713 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∃wrex 2509 {crab 2512 ∖ cdif 3194 ifcif 3602 {csn 3666 {cpr 3667 class class class wbr 4083 ↦ cmpt 4145 ‘cfv 5321 (class class class)co 6010 0cc0 8015 1c1 8016 · cmul 8020 < clt 8197 − cmin 8333 -cneg 8334 / cdiv 8835 ℕcn 9126 2c2 9177 4c4 9179 7c7 9182 8c8 9183 ℕ₀cn0 9385 ℤcz 9462 ...cfz 10221 ⌊cfl 10505 mod cmo 10561 ↑cexp 10777 ♯chash 11014 ∥ cdvds 12319 ℙcprime 12650 /_L clgs 15697

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-xor 1418 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-1st 6295 df-2nd 6296 df-recs 6462 df-irdg 6527 df-frec 6548 df-1o 6573 df-2o 6574 df-oadd 6577 df-er 6693 df-en 6901 df-dom 6902 df-fin 6903 df-sup 7167 df-inf 7168 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-5 9188 df-6 9189 df-7 9190 df-8 9191 df-n0 9386 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-ioo 10105 df-ico 10107 df-fz 10222 df-fzo 10356 df-fl 10507 df-mod 10562 df-seqfrec 10687 df-exp 10778 df-fac 10965 df-ihash 11015 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-clim 11811 df-proddc 12083 df-dvds 12320 df-gcd 12496 df-prm 12651 df-phi 12754 df-pc 12829 df-lgs 15698

This theorem is referenced by: 2lgsoddprm 15813

W3C validator