irrapxlem1 - Mathbox for Stefan O'Rear

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Theorem irrapxlem1 39412

Description: Lemma for irrapx1 39418. Divides the unit interval into 𝐵 half-open sections and using the pigeonhole principle fphpdo 39407 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 12942	. . . 4 ⊢ (0...𝐵) ⊆ (ℤ_≥‘0)
2		uzssz 12258	. . . . 5 ⊢ (ℤ_≥‘0) ⊆ ℤ
3		zssre 11982	. . . . 5 ⊢ ℤ ⊆ ℝ
4	2, 3	sstri 3975	. . . 4 ⊢ (ℤ_≥‘0) ⊆ ℝ
5	1, 4	sstri 3975	. . 3 ⊢ (0...𝐵) ⊆ ℝ
6	5	a1i 11	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...𝐵) ⊆ ℝ)
7		ovexd 7185	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (0...(𝐵 − 1)) ∈ V)
8		nnm1nn0 11932	. . . . 5 ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ ℕ₀)
9	8	adantl 484	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ ℕ₀)
10		nn0uz 12274	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11	9, 10	eleqtrdi 2923	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) ∈ (ℤ_≥‘0))
12		nnz 11998	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
13	12	adantl 484	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℤ)
14		nnre 11639	. . . . 5 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℝ)
15	14	adantl 484	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℝ)
16	15	ltm1d 11566	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (𝐵 − 1) < 𝐵)
17		fzsdom2 13783	. . 3 ⊢ ((((𝐵 − 1) ∈ (ℤ_≥‘0) ∧ 𝐵 ∈ ℤ) ∧ (𝐵 − 1) < 𝐵) → (0...(𝐵 − 1)) ≺ (0...𝐵))
18	11, 13, 16, 17	syl21anc 835	. 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 12391	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
21	20	ad2antrr 724	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐴 ∈ ℝ)
22		elfzelz 12902	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
23	22	zred 12081	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℝ)
24	23	adantl 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝑎 ∈ ℝ)
25	21, 24	remulcld 10665	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐴 · 𝑎) ∈ ℝ)
26		1rp 12387	. . . . . . 7 ⊢ 1 ∈ ℝ⁺
27		modcl 13235	. . . . . . 7 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
28	25, 26, 27	sylancl 588	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
29	19, 28	remulcld 10665	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) ∈ ℝ)
30	29	flcld 13162	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ)
31	19	recnd 10663	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 𝐵 ∈ ℂ)
32	31	mul01d 10833	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) = 0)
33		modge0 13241	. . . . . . . . . 10 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → 0 ≤ ((𝐴 · 𝑎) mod 1))
34	25, 26, 33	sylancl 588	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ ((𝐴 · 𝑎) mod 1))
35		0red 10638	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ∈ ℝ)
36		nngt0 11662	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℕ → 0 < 𝐵)
37	36	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 < 𝐵)
38		lemul2 11487	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ ((𝐴 · 𝑎) mod 1) ↔ (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1))))
40	34, 39	mpbid 234	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 0) ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
41	32, 40	eqbrtrrd 5082	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)))
42	35, 29	lenltd 10780	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (𝐵 · ((𝐴 · 𝑎) mod 1)) ↔ ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0))
43	41, 42	mpbid 234	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (𝐵 · ((𝐴 · 𝑎) mod 1)) < 0)
44		0z 11986	. . . . . . 7 ⊢ 0 ∈ ℤ
45		fllt 13170	. . . . . . 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 588	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐵 · ((𝐴 · 𝑎) mod 1)) < 0 ↔ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
47	43, 46	mtbid 326	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0)
48	30	zred 12081	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℝ)
49	35, 48	lenltd 10780	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ↔ ¬ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 0))
50	47, 49	mpbird 259	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))))
51		elnn0z 11988	. . . 4 ⊢ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℕ₀ ↔ ((⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ ℤ ∧ 0 ≤ (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1)))))
52	30, 50, 51	sylanbrc 585	. . 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 13163	. . . . . . 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 13242	. . . . . . . . 9 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ 1 ∈ ℝ⁺) → ((𝐴 · 𝑎) mod 1) < 1)
57	25, 26, 56	sylancl 588	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → ((𝐴 · 𝑎) mod 1) < 1)
58		1red 10636	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → 1 ∈ ℝ)
59		ltmul2 11485	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (((𝐴 · 𝑎) mod 1) < 1 ↔ (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1)))
61	57, 60	mpbid 234	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < (𝐵 · 1))
62	31	mulid1d 10652	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · 1) = 𝐵)
63	61, 62	breqtrd 5084	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 · ((𝐴 · 𝑎) mod 1)) < 𝐵)
64	48, 29, 19, 55, 63	lelttrd 10792	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) < 𝐵)
65		nncn 11640	. . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ)
66		ax-1cn 10589	. . . . . . 7 ⊢ 1 ∈ ℂ
67		npcan 10889	. . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵)
68	65, 66, 67	sylancl 588	. . . . . 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 5086	. . . 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 12006	. . . . . 6 ⊢ 1 ∈ ℤ
73		zsubcl 12018	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐵 − 1) ∈ ℤ)
74	71, 72, 73	sylancl 588	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (𝐵 − 1) ∈ ℤ)
75		zleltp1 12027	. . . . 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 586	. . . 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 259	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ≤ (𝐵 − 1))
78		elfz2nn0 12992	. . 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 1339	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (0...𝐵)) → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) ∈ (0...(𝐵 − 1)))
80		oveq2 7158	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐴 · 𝑎) = (𝐴 · 𝑥))
81	80	oveq1d 7165	. . . 4 ⊢ (𝑎 = 𝑥 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑥) mod 1))
82	81	oveq2d 7166	. . 3 ⊢ (𝑎 = 𝑥 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑥) mod 1)))
83	82	fveq2d 6668	. 2 ⊢ (𝑎 = 𝑥 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑥) mod 1))))
84		oveq2 7158	. . . . 5 ⊢ (𝑎 = 𝑦 → (𝐴 · 𝑎) = (𝐴 · 𝑦))
85	84	oveq1d 7165	. . . 4 ⊢ (𝑎 = 𝑦 → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑦) mod 1))
86	85	oveq2d 7166	. . 3 ⊢ (𝑎 = 𝑦 → (𝐵 · ((𝐴 · 𝑎) mod 1)) = (𝐵 · ((𝐴 · 𝑦) mod 1)))
87	86	fveq2d 6668	. 2 ⊢ (𝑎 = 𝑦 → (⌊‘(𝐵 · ((𝐴 · 𝑎) mod 1))) = (⌊‘(𝐵 · ((𝐴 · 𝑦) mod 1))))
88	6, 7, 18, 79, 83, 87	fphpdo 39407	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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ≺ csdm 8502 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383 ...cfz 12886 ⌊cfl 13154 mod cmo 13231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fl 13156 df-mod 13232 df-hash 13685

This theorem is referenced by: irrapxlem2 39413

W3C validator