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Theorem odd2np1lem 15681

Description: Lemma for odd2np1 15682. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

**Assertion**
Ref	Expression
odd2np1lem	⊢ (𝑁 ∈ ℕ₀ → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁))

Distinct variable groups: 𝑘,𝑁 𝑛,𝑁

Proof of Theorem odd2np1lem Dummy variables 𝑗 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2810	. . . 4 ⊢ (𝑗 = 0 → (((2 · 𝑛) + 1) = 𝑗 ↔ ((2 · 𝑛) + 1) = 0))
2	1	rexbidv 3256	. . 3 ⊢ (𝑗 = 0 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 0))
3		eqeq2 2810	. . . 4 ⊢ (𝑗 = 0 → ((𝑘 · 2) = 𝑗 ↔ (𝑘 · 2) = 0))
4	3	rexbidv 3256	. . 3 ⊢ (𝑗 = 0 → (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗 ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = 0))
5	2, 4	orbi12d 916	. 2 ⊢ (𝑗 = 0 → ((∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗) ↔ (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 0 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 0)))
6		eqeq2 2810	. . . . 5 ⊢ (𝑗 = 𝑚 → (((2 · 𝑛) + 1) = 𝑗 ↔ ((2 · 𝑛) + 1) = 𝑚))
7	6	rexbidv 3256	. . . 4 ⊢ (𝑗 = 𝑚 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑚))
8		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = 𝑥 → (2 · 𝑛) = (2 · 𝑥))
9	8	oveq1d 7150	. . . . . 6 ⊢ (𝑛 = 𝑥 → ((2 · 𝑛) + 1) = ((2 · 𝑥) + 1))
10	9	eqeq1d 2800	. . . . 5 ⊢ (𝑛 = 𝑥 → (((2 · 𝑛) + 1) = 𝑚 ↔ ((2 · 𝑥) + 1) = 𝑚))
11	10	cbvrexvw 3397	. . . 4 ⊢ (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑚 ↔ ∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚)
12	7, 11	syl6bb 290	. . 3 ⊢ (𝑗 = 𝑚 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ↔ ∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚))
13		eqeq2 2810	. . . . 5 ⊢ (𝑗 = 𝑚 → ((𝑘 · 2) = 𝑗 ↔ (𝑘 · 2) = 𝑚))
14	13	rexbidv 3256	. . . 4 ⊢ (𝑗 = 𝑚 → (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗 ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑚))
15		oveq1 7142	. . . . . 6 ⊢ (𝑘 = 𝑦 → (𝑘 · 2) = (𝑦 · 2))
16	15	eqeq1d 2800	. . . . 5 ⊢ (𝑘 = 𝑦 → ((𝑘 · 2) = 𝑚 ↔ (𝑦 · 2) = 𝑚))
17	16	cbvrexvw 3397	. . . 4 ⊢ (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑚 ↔ ∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚)
18	14, 17	syl6bb 290	. . 3 ⊢ (𝑗 = 𝑚 → (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗 ↔ ∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚))
19	12, 18	orbi12d 916	. 2 ⊢ (𝑗 = 𝑚 → ((∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗) ↔ (∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚 ∨ ∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚)))
20		eqeq2 2810	. . . 4 ⊢ (𝑗 = (𝑚 + 1) → (((2 · 𝑛) + 1) = 𝑗 ↔ ((2 · 𝑛) + 1) = (𝑚 + 1)))
21	20	rexbidv 3256	. . 3 ⊢ (𝑗 = (𝑚 + 1) → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
22		eqeq2 2810	. . . 4 ⊢ (𝑗 = (𝑚 + 1) → ((𝑘 · 2) = 𝑗 ↔ (𝑘 · 2) = (𝑚 + 1)))
23	22	rexbidv 3256	. . 3 ⊢ (𝑗 = (𝑚 + 1) → (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗 ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1)))
24	21, 23	orbi12d 916	. 2 ⊢ (𝑗 = (𝑚 + 1) → ((∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗) ↔ (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1) ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1))))
25		eqeq2 2810	. . . 4 ⊢ (𝑗 = 𝑁 → (((2 · 𝑛) + 1) = 𝑗 ↔ ((2 · 𝑛) + 1) = 𝑁))
26	25	rexbidv 3256	. . 3 ⊢ (𝑗 = 𝑁 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁))
27		eqeq2 2810	. . . 4 ⊢ (𝑗 = 𝑁 → ((𝑘 · 2) = 𝑗 ↔ (𝑘 · 2) = 𝑁))
28	27	rexbidv 3256	. . 3 ⊢ (𝑗 = 𝑁 → (∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗 ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁))
29	26, 28	orbi12d 916	. 2 ⊢ (𝑗 = 𝑁 → ((∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑗 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑗) ↔ (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁)))
30		0z 11980	. . . 4 ⊢ 0 ∈ ℤ
31		2cn 11700	. . . . 5 ⊢ 2 ∈ ℂ
32	31	mul02i 10818	. . . 4 ⊢ (0 · 2) = 0
33		oveq1 7142	. . . . . 6 ⊢ (𝑘 = 0 → (𝑘 · 2) = (0 · 2))
34	33	eqeq1d 2800	. . . . 5 ⊢ (𝑘 = 0 → ((𝑘 · 2) = 0 ↔ (0 · 2) = 0))
35	34	rspcev 3571	. . . 4 ⊢ ((0 ∈ ℤ ∧ (0 · 2) = 0) → ∃𝑘 ∈ ℤ (𝑘 · 2) = 0)
36	30, 32, 35	mp2an 691	. . 3 ⊢ ∃𝑘 ∈ ℤ (𝑘 · 2) = 0
37	36	olci 863	. 2 ⊢ (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 0 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 0)
38		orcom 867	. . 3 ⊢ ((∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚 ∨ ∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚) ↔ (∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚 ∨ ∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚))
39		zcn 11974	. . . . . . . . 9 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
40		mulcom 10612	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℂ ∧ 2 ∈ ℂ) → (𝑦 · 2) = (2 · 𝑦))
41	39, 31, 40	sylancl 589	. . . . . . . 8 ⊢ (𝑦 ∈ ℤ → (𝑦 · 2) = (2 · 𝑦))
42	41	adantl 485	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑦 ∈ ℤ) → (𝑦 · 2) = (2 · 𝑦))
43	42	eqeq1d 2800	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑦 ∈ ℤ) → ((𝑦 · 2) = 𝑚 ↔ (2 · 𝑦) = 𝑚))
44		eqid 2798	. . . . . . . . 9 ⊢ ((2 · 𝑦) + 1) = ((2 · 𝑦) + 1)
45		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑦 → (2 · 𝑛) = (2 · 𝑦))
46	45	oveq1d 7150	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑦 → ((2 · 𝑛) + 1) = ((2 · 𝑦) + 1))
47	46	eqeq1d 2800	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (((2 · 𝑛) + 1) = ((2 · 𝑦) + 1) ↔ ((2 · 𝑦) + 1) = ((2 · 𝑦) + 1)))
48	47	rspcev 3571	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤ ∧ ((2 · 𝑦) + 1) = ((2 · 𝑦) + 1)) → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = ((2 · 𝑦) + 1))
49	44, 48	mpan2 690	. . . . . . . 8 ⊢ (𝑦 ∈ ℤ → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = ((2 · 𝑦) + 1))
50		oveq1 7142	. . . . . . . . . 10 ⊢ ((2 · 𝑦) = 𝑚 → ((2 · 𝑦) + 1) = (𝑚 + 1))
51	50	eqeq2d 2809	. . . . . . . . 9 ⊢ ((2 · 𝑦) = 𝑚 → (((2 · 𝑛) + 1) = ((2 · 𝑦) + 1) ↔ ((2 · 𝑛) + 1) = (𝑚 + 1)))
52	51	rexbidv 3256	. . . . . . . 8 ⊢ ((2 · 𝑦) = 𝑚 → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = ((2 · 𝑦) + 1) ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
53	49, 52	syl5ibcom 248	. . . . . . 7 ⊢ (𝑦 ∈ ℤ → ((2 · 𝑦) = 𝑚 → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
54	53	adantl 485	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑦 ∈ ℤ) → ((2 · 𝑦) = 𝑚 → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
55	43, 54	sylbid 243	. . . . 5 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑦 ∈ ℤ) → ((𝑦 · 2) = 𝑚 → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
56	55	rexlimdva 3243	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚 → ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1)))
57		peano2z 12011	. . . . . . 7 ⊢ (𝑥 ∈ ℤ → (𝑥 + 1) ∈ ℤ)
58		zcn 11974	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
59		mulcom 10612	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 2 ∈ ℂ) → (𝑥 · 2) = (2 · 𝑥))
60	31, 59	mpan2 690	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (𝑥 · 2) = (2 · 𝑥))
61	31	mulid2i 10635	. . . . . . . . . . . . 13 ⊢ (1 · 2) = 2
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (1 · 2) = 2)
63	60, 62	oveq12d 7153	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → ((𝑥 · 2) + (1 · 2)) = ((2 · 𝑥) + 2))
64		df-2 11688	. . . . . . . . . . . 12 ⊢ 2 = (1 + 1)
65	64	oveq2i 7146	. . . . . . . . . . 11 ⊢ ((2 · 𝑥) + 2) = ((2 · 𝑥) + (1 + 1))
66	63, 65	eqtrdi 2849	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑥 · 2) + (1 · 2)) = ((2 · 𝑥) + (1 + 1)))
67		ax-1cn 10584	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
68		adddir 10621	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 1 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑥 + 1) · 2) = ((𝑥 · 2) + (1 · 2)))
69	67, 31, 68	mp3an23 1450	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → ((𝑥 + 1) · 2) = ((𝑥 · 2) + (1 · 2)))
70		mulcl 10610	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (2 · 𝑥) ∈ ℂ)
71	31, 70	mpan 689	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (2 · 𝑥) ∈ ℂ)
72		addass 10613	. . . . . . . . . . . 12 ⊢ (((2 · 𝑥) ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ∈ ℂ) → (((2 · 𝑥) + 1) + 1) = ((2 · 𝑥) + (1 + 1)))
73	67, 67, 72	mp3an23 1450	. . . . . . . . . . 11 ⊢ ((2 · 𝑥) ∈ ℂ → (((2 · 𝑥) + 1) + 1) = ((2 · 𝑥) + (1 + 1)))
74	71, 73	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (((2 · 𝑥) + 1) + 1) = ((2 · 𝑥) + (1 + 1)))
75	66, 69, 74	3eqtr4d 2843	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((𝑥 + 1) · 2) = (((2 · 𝑥) + 1) + 1))
76	58, 75	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℤ → ((𝑥 + 1) · 2) = (((2 · 𝑥) + 1) + 1))
77	76	adantl 485	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑥 ∈ ℤ) → ((𝑥 + 1) · 2) = (((2 · 𝑥) + 1) + 1))
78		oveq1 7142	. . . . . . . . 9 ⊢ (𝑘 = (𝑥 + 1) → (𝑘 · 2) = ((𝑥 + 1) · 2))
79	78	eqeq1d 2800	. . . . . . . 8 ⊢ (𝑘 = (𝑥 + 1) → ((𝑘 · 2) = (((2 · 𝑥) + 1) + 1) ↔ ((𝑥 + 1) · 2) = (((2 · 𝑥) + 1) + 1)))
80	79	rspcev 3571	. . . . . . 7 ⊢ (((𝑥 + 1) ∈ ℤ ∧ ((𝑥 + 1) · 2) = (((2 · 𝑥) + 1) + 1)) → ∃𝑘 ∈ ℤ (𝑘 · 2) = (((2 · 𝑥) + 1) + 1))
81	57, 77, 80	syl2an2 685	. . . . . 6 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑥 ∈ ℤ) → ∃𝑘 ∈ ℤ (𝑘 · 2) = (((2 · 𝑥) + 1) + 1))
82		oveq1 7142	. . . . . . . 8 ⊢ (((2 · 𝑥) + 1) = 𝑚 → (((2 · 𝑥) + 1) + 1) = (𝑚 + 1))
83	82	eqeq2d 2809	. . . . . . 7 ⊢ (((2 · 𝑥) + 1) = 𝑚 → ((𝑘 · 2) = (((2 · 𝑥) + 1) + 1) ↔ (𝑘 · 2) = (𝑚 + 1)))
84	83	rexbidv 3256	. . . . . 6 ⊢ (((2 · 𝑥) + 1) = 𝑚 → (∃𝑘 ∈ ℤ (𝑘 · 2) = (((2 · 𝑥) + 1) + 1) ↔ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1)))
85	81, 84	syl5ibcom 248	. . . . 5 ⊢ ((𝑚 ∈ ℕ₀ ∧ 𝑥 ∈ ℤ) → (((2 · 𝑥) + 1) = 𝑚 → ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1)))
86	85	rexlimdva 3243	. . . 4 ⊢ (𝑚 ∈ ℕ₀ → (∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚 → ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1)))
87	56, 86	orim12d 962	. . 3 ⊢ (𝑚 ∈ ℕ₀ → ((∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚 ∨ ∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚) → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1) ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1))))
88	38, 87	syl5bi 245	. 2 ⊢ (𝑚 ∈ ℕ₀ → ((∃𝑥 ∈ ℤ ((2 · 𝑥) + 1) = 𝑚 ∨ ∃𝑦 ∈ ℤ (𝑦 · 2) = 𝑚) → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = (𝑚 + 1) ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = (𝑚 + 1))))
89	5, 19, 24, 29, 37, 88	nn0ind 12065	1 ⊢ (𝑁 ∈ ℕ₀ → (∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁 ∨ ∃𝑘 ∈ ℤ (𝑘 · 2) = 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 2c2 11680 ℕ₀cn0 11885 ℤcz 11969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970

This theorem is referenced by: odd2np1 15682

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