2lgs - Intuitionistic Logic Explorer

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Theorem 2lgs 15826

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 15739) 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 12691	. 2 ⊢ (𝑃 ∈ ℙ → (𝑃 = 2 ∨ ¬ 2 ∥ 𝑃))
2		2lgslem4 15825	. . . . . 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 6021	. . . . . 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 6020	. . . . . 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 12692	. . . . . . . . . 10 ⊢ 2 ∈ ℙ
11		prmnn 12675	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
12		dvdsprime 12687	. . . . . . . . . 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 12468	. . . . . . . . . . . . 13 ⊢ 2 ∥ 2
15		breq2 4090	. . . . . . . . . . . . 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 12679	. . . . . . . . . . . . 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 9500	. . . . . . 7 ⊢ 2 ∈ ℤ
27	25, 26	jctil 312	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (2 ∈ ℤ ∧ ¬ 𝑃 ∥ 2))
28		2lgslem1 15813	. . . . . . 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 12828	. . . . . . . . . 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 15791	. . . . . . 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 15814	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ ¬ 2 ∥ 𝑃) → (((𝑃 − 1) / 2) − (⌊‘(𝑃 / 4))) ∈ ℤ)
41		m1exp1 12455	. . . . . 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 9298	. . . . . . 7 ⊢ 2 ∈ ℕ
44		dvdsval3 12345	. . . . . . 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 15823	. . . . . . . 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 12676	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
50		8nn 9304	. . . . . . . . . . . . . . . 16 ⊢ 8 ∈ ℕ
51	50	a1i 9	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 8 ∈ ℕ)
52	49, 51	zmodcld 10600	. . . . . . . . . . . . . 14 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℕ₀)
53	52	nn0zd 9593	. . . . . . . . . . . . 13 ⊢ (𝑃 ∈ ℙ → (𝑃 mod 8) ∈ ℤ)
54		1z 9498	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
55		zdceq 9548	. . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . 14 ⊢ 7 ∈ ℕ
58	57	nnzi 9493	. . . . . . . . . . . . 13 ⊢ 7 ∈ ℤ
59		zdceq 9548	. . . . . . . . . . . . 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 3687	. . . . . . . . . . . . 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 3611	. . . . . . . . . . . 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 9204	. . . . . . . . . . . 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 3608	. . . . . . . 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 3195 ifcif 3603 {csn 3667 {cpr 3668 class class class wbr 4086 ↦ cmpt 4148 ‘cfv 5324 (class class class)co 6013 0cc0 8025 1c1 8026 · cmul 8030 < clt 8207 − cmin 8343 -cneg 8344 / cdiv 8845 ℕcn 9136 2c2 9187 4c4 9189 7c7 9192 8c8 9193 ℕ₀cn0 9395 ℤcz 9472 ...cfz 10236 ⌊cfl 10521 mod cmo 10577 ↑cexp 10793 ♯chash 11030 ∥ cdvds 12341 ℙcprime 12672 /_L clgs 15719

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 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

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 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-tp 3675 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-2o 6578 df-oadd 6581 df-er 6697 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7177 df-inf 7178 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-5 9198 df-6 9199 df-7 9200 df-8 9201 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-ioo 10120 df-ico 10122 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-fac 10981 df-ihash 11031 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-clim 11833 df-proddc 12105 df-dvds 12342 df-gcd 12518 df-prm 12673 df-phi 12776 df-pc 12851 df-lgs 15720

This theorem is referenced by: 2lgsoddprm 15835

W3C validator