irrapxlem1 - Mathbox for Stefan O'Rear

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Theorem irrapxlem1 43607

Description: Lemma for irrapx1 43613. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 43602 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 13610	. . . 4 ⊢ (0...𝐵) ⊆ (ℤ_≥‘0)
2		uzssz 12899	. . . . 5 ⊢ (ℤ_≥‘0) ⊆ ℤ
3		zssre 12613	. . . . 5 ⊢ ℤ ⊆ ℝ
4	2, 3	sstri 3947	. . . 4 ⊢ (ℤ_≥‘0) ⊆ ℝ
5	1, 4	sstri 3947	. . 3 ⊢ (0...𝐵) ⊆ ℝ
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...𝐵) ⊆ ℝ)
7		ovexd 7454	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ∈ V)
8		nnm1nn0 12560	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ ℕ₀)
9	8	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ ℕ₀)
10		nn0uz 12916	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2875	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ (ℤ_≥‘0))
12		nnz 12627	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
13	12	adantl 487	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ)
14		nnre 12255	. . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
15	14	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ)
16	15	ltm1d 12162	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) < 𝐵)
17		fzsdom2 14483	. . 3 ⊢ ((((𝐵 − 1) ∈ (ℤ_≥‘0) ∧ 𝐵 ∈ ℤ) ∧ (𝐵 − 1) < 𝐵) → (0...(𝐵 − 1)) ≺ (0...𝐵))
18	11, 13, 16, 17	syl21anc 851	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ≺ (0...𝐵))
19	14	ad2antlr 740	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℝ)
20		rpre 13041	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
21	20	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐴 ∈ ℝ)
22		elfzelz 13568	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
23	22	zred 12716	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℝ)
24	23	adantl 487	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝑎 ∈ ℝ)
25	21, 24	remulcld 11254	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐴 · 𝑎) ∈ ℝ)
26		1rp 13036	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
27		modcl 13924	. . . . . . 7 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
28	25, 26, 27	sylancl 598	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
29	19, 28	remulcld 11254	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ)
30	29	flcld 13849	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ)
31	19	recnd 11252	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℂ)
32	31	mul01d 11424	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) = 0)
33		modge0 13930	. . . . . . . . . 10 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → 0 ≤ ((𝐴 · 𝑎) mod 1))
34	25, 26, 33	sylancl 598	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ ((𝐴 · 𝑎) mod 1))
35		0red 11226	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ∈ ℝ)
36		nngt0 12282	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
37	36	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 < 𝐵)
38		lemul2 12083	. . . . . . . . . 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 1401	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ ((𝐴 · 𝑎) mod 1) ↔ (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1))))
40	34, 39	mpbid 235	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
41	32, 40	eqbrtrrd 5137	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
42	35, 29	lenltd 11371	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)) ↔ ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0))
43	41, 42	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0)
44		0z 12617	. . . . . . 7 ⊢ 0 ∈ ℤ
45		fllt 13857	. . . . . . 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 598	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 · ((𝐴 · 𝑎) mod 1)) < 0 ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
47	43, 46	mtbid 327	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0)
48	30	zred 12716	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℝ)
49	35, 48	lenltd 11371	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ↔ ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
50	47, 49	mpbird 260	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))))
51		elnn0z 12619	. . . 4 ⊢ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀ ↔ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ ∧ 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1)))))
52	30, 50, 51	sylanbrc 595	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀)
53	8	ad2antlr 740	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℕ₀)
54		flle 13850	. . . . . . 7 ⊢ ((𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
55	29, 54	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
56		modlt 13931	. . . . . . . . 9 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) < 1)
57	25, 26, 56	sylancl 598	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) < 1)
58		1red 11224	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 1 ∈ ℝ)
59		ltmul2 12081	. . . . . . . . 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 1401	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (((𝐴 · 𝑎) mod 1) < 1 ↔ (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1)))
61	57, 60	mpbid 235	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1))
62	31	mulridd 11241	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 1) = 𝐵)
63	61, 62	breqtrd 5139	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < 𝐵)
64	48, 29, 19, 55, 63	lelttrd 11383	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 𝐵)
65		nncn 12256	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
66		ax-1cn 11173	. . . . . . 7 ⊢ 1 ∈ ℂ
67		npcan 11481	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵)
68	65, 66, 67	sylancl 598	. . . . . 6 ⊢ (𝐵 ∈ ℕ → ((𝐵 − 1) + 1) = 𝐵)
69	68	ad2antlr 740	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 − 1) + 1) = 𝐵)
70	64, 69	breqtrrd 5141	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < ((𝐵 − 1) + 1))
71	12	ad2antlr 740	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℤ)
72		1z 12639	. . . . . 6 ⊢ 1 ∈ ℤ
73		zsubcl 12651	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐵 − 1) ∈ ℤ)
74	71, 72, 73	sylancl 598	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℤ)
75		zleltp1 12660	. . . . 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 596	. . . 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 260	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1))
78		elfz2nn0 13663	. . 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 1362	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ (0...(𝐵 − 1)))
80		oveq2 7427	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐴 · 𝑎) = (𝐴 · 𝑥))
81	80	oveq1d 7434	. . . 4 ⊢ (𝑎 = 𝑥 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑥) mod 1))
82	81	oveq2d 7435	. . 3 ⊢ (𝑎 = 𝑥 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑥) mod 1)))
83	82	fveq2d 6889	. 2 ⊢ (𝑎 = 𝑥 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))))
84		oveq2 7427	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝐴 · 𝑎) = (𝐴 · 𝑦))
85	84	oveq1d 7434	. . . 4 ⊢ (𝑎 = 𝑦 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑦) mod 1))
86	85	oveq2d 7435	. . 3 ⊢ (𝑎 = 𝑦 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑦) mod 1)))
87	86	fveq2d 6889	. 2 ⊢ (𝑎 = 𝑦 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))
88	6, 7, 18, 79, 83, 87	fphpdo 43602	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∃wrex 3091 ⊆ wss 3906 class class class wbr 5111 ‘cfv 6540 (class class class)co 7419 ≺ csdm 8948 ℂcc 11113 ℝcr 11114 0cc0 11115 1c1 11116 + caddc 11118 · cmul 11120 < clt 11258 ≤ cle 11259 − cmin 11456 ℕcn 12248 ℕ₀cn0 12519 ℤcz 12606 ℤ_≥cuz 12878 ℝ⁺crp 13032 ...cfz 13551 ⌊cfl 13841 mod cmo 13920

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-n0 12520 df-xnn0 12593 df-z 12607 df-uz 12879 df-rp 13033 df-fz 13552 df-fl 13843 df-mod 13921 df-hash 14385

This theorem is used by: irrapxlem2 43608

W3C validator