irrapxlem1 - Mathbox for Stefan O'Rear

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Theorem irrapxlem1 43179

Description: Lemma for irrapx1 43185. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 43174 finds two multiples of 𝐴 in the same section mod 1. (Contributed by Stefan O'Rear, 12-Sep-2014.)

**Assertion**
Ref	Expression
irrapxlem1	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1)))))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

Proof of Theorem irrapxlem1 Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzssuz 13493	. . . 4 ⊢ (0...𝐵) ⊆ (ℤ_≥‘0)
2		uzssz 12784	. . . . 5 ⊢ (ℤ_≥‘0) ⊆ ℤ
3		zssre 12507	. . . . 5 ⊢ ℤ ⊆ ℝ
4	2, 3	sstri 3945	. . . 4 ⊢ (ℤ_≥‘0) ⊆ ℝ
5	1, 4	sstri 3945	. . 3 ⊢ (0...𝐵) ⊆ ℝ
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...𝐵) ⊆ ℝ)
7		ovexd 7403	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ∈ V)
8		nnm1nn0 12454	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ ℕ₀)
9	8	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ ℕ₀)
10		nn0uz 12801	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2847	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ (ℤ_≥‘0))
12		nnz 12521	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
13	12	adantl 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ)
14		nnre 12164	. . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
15	14	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ)
16	15	ltm1d 12086	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) < 𝐵)
17		fzsdom2 14363	. . 3 ⊢ ((((𝐵 − 1) ∈ (ℤ_≥‘0) ∧ 𝐵 ∈ ℤ) ∧ (𝐵 − 1) < 𝐵) → (0...(𝐵 − 1)) ≺ (0...𝐵))
18	11, 13, 16, 17	syl21anc 838	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ≺ (0...𝐵))
19	14	ad2antlr 728	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℝ)
20		rpre 12926	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
21	20	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐴 ∈ ℝ)
22		elfzelz 13452	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
23	22	zred 12608	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℝ)
24	23	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝑎 ∈ ℝ)
25	21, 24	remulcld 11174	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐴 · 𝑎) ∈ ℝ)
26		1rp 12921	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
27		modcl 13805	. . . . . . 7 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
28	25, 26, 27	sylancl 587	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
29	19, 28	remulcld 11174	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ)
30	29	flcld 13730	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ)
31	19	recnd 11172	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℂ)
32	31	mul01d 11344	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) = 0)
33		modge0 13811	. . . . . . . . . 10 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → 0 ≤ ((𝐴 · 𝑎) mod 1))
34	25, 26, 33	sylancl 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ ((𝐴 · 𝑎) mod 1))
35		0red 11147	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ∈ ℝ)
36		nngt0 12188	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
37	36	ad2antlr 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 < 𝐵)
38		lemul2 12006	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ ((𝐴 · 𝑎) mod 1) ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (0 ≤ ((𝐴 · 𝑎) mod 1) ↔ (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1))))
39	35, 28, 19, 37, 38	syl112anc 1377	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ ((𝐴 · 𝑎) mod 1) ↔ (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1))))
40	34, 39	mpbid 232	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
41	32, 40	eqbrtrrd 5124	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
42	35, 29	lenltd 11291	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)) ↔ ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0))
43	41, 42	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0)
44		0z 12511	. . . . . . 7 ⊢ 0 ∈ ℤ
45		fllt 13738	. . . . . . 7 ⊢ (((𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ ∧ 0 ∈ ℤ) → ((𝐵 · ((𝐴 · 𝑎) mod 1)) < 0 ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
46	29, 44, 45	sylancl 587	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 · ((𝐴 · 𝑎) mod 1)) < 0 ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
47	43, 46	mtbid 324	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0)
48	30	zred 12608	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℝ)
49	35, 48	lenltd 11291	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ↔ ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
50	47, 49	mpbird 257	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))))
51		elnn0z 12513	. . . 4 ⊢ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀ ↔ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ ∧ 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1)))))
52	30, 50, 51	sylanbrc 584	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀)
53	8	ad2antlr 728	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℕ₀)
54		flle 13731	. . . . . . 7 ⊢ ((𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
55	29, 54	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
56		modlt 13812	. . . . . . . . 9 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) < 1)
57	25, 26, 56	sylancl 587	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) < 1)
58		1red 11145	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 1 ∈ ℝ)
59		ltmul2 12004	. . . . . . . . 9 ⊢ ((((𝐴 · 𝑎) mod 1) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (((𝐴 · 𝑎) mod 1) < 1 ↔ (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1)))
60	28, 58, 19, 37, 59	syl112anc 1377	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (((𝐴 · 𝑎) mod 1) < 1 ↔ (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1)))
61	57, 60	mpbid 232	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1))
62	31	mulridd 11161	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 1) = 𝐵)
63	61, 62	breqtrd 5126	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < 𝐵)
64	48, 29, 19, 55, 63	lelttrd 11303	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 𝐵)
65		nncn 12165	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
66		ax-1cn 11096	. . . . . . 7 ⊢ 1 ∈ ℂ
67		npcan 11401	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵)
68	65, 66, 67	sylancl 587	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((𝐵 − 1) + 1) = 𝐵)
69	68	ad2antlr 728	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 − 1) + 1) = 𝐵)
70	64, 69	breqtrrd 5128	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < ((𝐵 − 1) + 1))
71	12	ad2antlr 728	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℤ)
72		1z 12533	. . . . . 6 ⊢ 1 ∈ ℤ
73		zsubcl 12545	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐵 − 1) ∈ ℤ)
74	71, 72, 73	sylancl 587	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℤ)
75		zleltp1 12554	. . . . 5 ⊢ (((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ ∧ (𝐵 − 1) ∈ ℤ) → ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1) ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < ((𝐵 − 1) + 1)))
76	30, 74, 75	syl2anc 585	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1) ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < ((𝐵 − 1) + 1)))
77	70, 76	mpbird 257	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1))
78		elfz2nn0 13546	. . 3 ⊢ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ (0...(𝐵 − 1)) ↔ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀ ∧ (𝐵 − 1) ∈ ℕ₀ ∧ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1)))
79	52, 53, 77, 78	syl3anbrc 1345	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ (0...(𝐵 − 1)))
80		oveq2 7376	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐴 · 𝑎) = (𝐴 · 𝑥))
81	80	oveq1d 7383	. . . 4 ⊢ (𝑎 = 𝑥 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑥) mod 1))
82	81	oveq2d 7384	. . 3 ⊢ (𝑎 = 𝑥 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑥) mod 1)))
83	82	fveq2d 6846	. 2 ⊢ (𝑎 = 𝑥 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))))
84		oveq2 7376	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝐴 · 𝑎) = (𝐴 · 𝑦))
85	84	oveq1d 7383	. . . 4 ⊢ (𝑎 = 𝑦 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑦) mod 1))
86	85	oveq2d 7384	. . 3 ⊢ (𝑎 = 𝑦 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑦) mod 1)))
87	86	fveq2d 6846	. 2 ⊢ (𝑎 = 𝑦 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))
88	6, 7, 18, 79, 83, 87	fphpdo 43174	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (0...𝐵)∃𝑦 ∈ (0...𝐵)(𝑥 < 𝑦 ∧ (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 ⊆ wss 3903 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ≺ csdm 8894 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ℝ⁺crp 12917 ...cfz 13435 ⌊cfl 13722 mod cmo 13801

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-fz 13436 df-fl 13724 df-mod 13802 df-hash 14266

This theorem is referenced by: irrapxlem2 43180

W3C validator