irrapxlem1 - Mathbox for Stefan O'Rear

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Theorem irrapxlem1 42306

Description: Lemma for irrapx1 42312. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 42301 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 13572	. . . 4 ⊢ (0...𝐵) ⊆ (ℤ_≥‘0)
2		uzssz 12871	. . . . 5 ⊢ (ℤ_≥‘0) ⊆ ℤ
3		zssre 12593	. . . . 5 ⊢ ℤ ⊆ ℝ
4	2, 3	sstri 3982	. . . 4 ⊢ (ℤ_≥‘0) ⊆ ℝ
5	1, 4	sstri 3982	. . 3 ⊢ (0...𝐵) ⊆ ℝ
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...𝐵) ⊆ ℝ)
7		ovexd 7450	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ∈ V)
8		nnm1nn0 12541	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ ℕ₀)
9	8	adantl 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ ℕ₀)
10		nn0uz 12892	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2835	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ (ℤ_≥‘0))
12		nnz 12607	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
13	12	adantl 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ)
14		nnre 12247	. . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
15	14	adantl 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ)
16	15	ltm1d 12174	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) < 𝐵)
17		fzsdom2 14417	. . 3 ⊢ ((((𝐵 − 1) ∈ (ℤ_≥‘0) ∧ 𝐵 ∈ ℤ) ∧ (𝐵 − 1) < 𝐵) → (0...(𝐵 − 1)) ≺ (0...𝐵))
18	11, 13, 16, 17	syl21anc 836	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ≺ (0...𝐵))
19	14	ad2antlr 725	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℝ)
20		rpre 13012	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
21	20	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐴 ∈ ℝ)
22		elfzelz 13531	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
23	22	zred 12694	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℝ)
24	23	adantl 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝑎 ∈ ℝ)
25	21, 24	remulcld 11272	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐴 · 𝑎) ∈ ℝ)
26		1rp 13008	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
27		modcl 13868	. . . . . . 7 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
28	25, 26, 27	sylancl 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
29	19, 28	remulcld 11272	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ)
30	29	flcld 13793	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ)
31	19	recnd 11270	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℂ)
32	31	mul01d 11441	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) = 0)
33		modge0 13874	. . . . . . . . . 10 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → 0 ≤ ((𝐴 · 𝑎) mod 1))
34	25, 26, 33	sylancl 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ ((𝐴 · 𝑎) mod 1))
35		0red 11245	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ∈ ℝ)
36		nngt0 12271	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
37	36	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 < 𝐵)
38		lemul2 12095	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ ((𝐴 · 𝑎) mod 1) ↔ (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1))))
40	34, 39	mpbid 231	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
41	32, 40	eqbrtrrd 5167	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
42	35, 29	lenltd 11388	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)) ↔ ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0))
43	41, 42	mpbid 231	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0)
44		0z 12597	. . . . . . 7 ⊢ 0 ∈ ℤ
45		fllt 13801	. . . . . . 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 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 · ((𝐴 · 𝑎) mod 1)) < 0 ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
47	43, 46	mtbid 323	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0)
48	30	zred 12694	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℝ)
49	35, 48	lenltd 11388	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ↔ ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
50	47, 49	mpbird 256	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))))
51		elnn0z 12599	. . . 4 ⊢ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀ ↔ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ ∧ 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1)))))
52	30, 50, 51	sylanbrc 581	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀)
53	8	ad2antlr 725	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℕ₀)
54		flle 13794	. . . . . . 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 13875	. . . . . . . . 9 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) < 1)
57	25, 26, 56	sylancl 584	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) < 1)
58		1red 11243	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 1 ∈ ℝ)
59		ltmul2 12093	. . . . . . . . 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 1371	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (((𝐴 · 𝑎) mod 1) < 1 ↔ (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1)))
61	57, 60	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1))
62	31	mulridd 11259	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 1) = 𝐵)
63	61, 62	breqtrd 5169	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < 𝐵)
64	48, 29, 19, 55, 63	lelttrd 11400	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 𝐵)
65		nncn 12248	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
66		ax-1cn 11194	. . . . . . 7 ⊢ 1 ∈ ℂ
67		npcan 11497	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵)
68	65, 66, 67	sylancl 584	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((𝐵 − 1) + 1) = 𝐵)
69	68	ad2antlr 725	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 − 1) + 1) = 𝐵)
70	64, 69	breqtrrd 5171	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < ((𝐵 − 1) + 1))
71	12	ad2antlr 725	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℤ)
72		1z 12620	. . . . . 6 ⊢ 1 ∈ ℤ
73		zsubcl 12632	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐵 − 1) ∈ ℤ)
74	71, 72, 73	sylancl 584	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℤ)
75		zleltp1 12641	. . . . 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 582	. . . 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 256	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1))
78		elfz2nn0 13622	. . 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 1340	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ (0...(𝐵 − 1)))
80		oveq2 7423	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐴 · 𝑎) = (𝐴 · 𝑥))
81	80	oveq1d 7430	. . . 4 ⊢ (𝑎 = 𝑥 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑥) mod 1))
82	81	oveq2d 7431	. . 3 ⊢ (𝑎 = 𝑥 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑥) mod 1)))
83	82	fveq2d 6895	. 2 ⊢ (𝑎 = 𝑥 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))))
84		oveq2 7423	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝐴 · 𝑎) = (𝐴 · 𝑦))
85	84	oveq1d 7430	. . . 4 ⊢ (𝑎 = 𝑦 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑦) mod 1))
86	85	oveq2d 7431	. . 3 ⊢ (𝑎 = 𝑦 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑦) mod 1)))
87	86	fveq2d 6895	. 2 ⊢ (𝑎 = 𝑦 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))
88	6, 7, 18, 79, 83, 87	fphpdo 42301	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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ⊆ wss 3940 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ≺ csdm 8959 ℂcc 11134 ℝcr 11135 0cc0 11136 1c1 11137 + caddc 11139 · cmul 11141 < clt 11276 ≤ cle 11277 − cmin 11472 ℕcn 12240 ℕ₀cn0 12500 ℤcz 12586 ℤ_≥cuz 12850 ℝ⁺crp 13004 ...cfz 13514 ⌊cfl 13785 mod cmo 13864

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-oadd 8487 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-fz 13515 df-fl 13787 df-mod 13865 df-hash 14320

This theorem is referenced by: irrapxlem2 42307

W3C validator