irrapxlem3 - Mathbox for Stefan O'Rear

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Theorem irrapxlem3 43401

Description: Lemma for irrapx1 43405. By subtraction, there is a multiple very close to an integer. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
irrapxlem3	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

Proof of Theorem irrapxlem3 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irrapxlem2 43400	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑎 ∈ (0...𝐵)∃𝑏 ∈ (0...𝐵)(𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵)))
2		1z 12601	. . . . . . . 8 ⊢ 1 ∈ ℤ
3	2	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ∈ ℤ)
4		simpllr 785	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℕ)
5	4	nnzd 12594	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℤ)
6		simplrr 787	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ (0...𝐵))
7	6	elfzelzd 13530	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℤ)
8		simplrl 786	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (0...𝐵))
9	8	elfzelzd 13530	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℤ)
10	7, 9	zsubcld 12682	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℤ)
11		1m1e0 12290	. . . . . . . . 9 ⊢ (1 − 1) = 0
12		elfzelz 13529	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
13	12	ad2antrl 738	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℤ)
14	13	zred 12677	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℝ)
15		elfzelz 13529	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (0...𝐵) → 𝑏 ∈ ℤ)
16	15	ad2antll 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℤ)
17	16	zred 12677	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℝ)
18	14, 17	posdifd 11774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → (𝑎 < 𝑏 ↔ 0 < (𝑏 − 𝑎)))
19	18	biimpa 480	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < (𝑏 − 𝑎))
20	11, 19	eqbrtrid 5135	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (1 − 1) < (𝑏 − 𝑎))
21		zlem1lt 12623	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (𝑏 − 𝑎) ∈ ℤ) → (1 ≤ (𝑏 − 𝑎) ↔ (1 − 1) < (𝑏 − 𝑎)))
22	2, 10, 21	sylancr 596	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (1 ≤ (𝑏 − 𝑎) ↔ (1 − 1) < (𝑏 − 𝑎)))
23	20, 22	mpbird 259	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ≤ (𝑏 − 𝑎))
24	7	zred 12677	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℝ)
25	9	zred 12677	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℝ)
26	24, 25	resubcld 11615	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℝ)
27		0red 11184	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 ∈ ℝ)
28	24, 27	resubcld 11615	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ∈ ℝ)
29	4	nnred 12225	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℝ)
30		elfzle1 13532	. . . . . . . . . 10 ⊢ (𝑎 ∈ (0...𝐵) → 0 ≤ 𝑎)
31	8, 30	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 ≤ 𝑎)
32	27, 25, 24, 31	lesub2dd 11804	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ (𝑏 − 0))
33	24	recnd 11210	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℂ)
34	33	subid1d 11531	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) = 𝑏)
35		elfzle2 13533	. . . . . . . . . 10 ⊢ (𝑏 ∈ (0...𝐵) → 𝑏 ≤ 𝐵)
36	6, 35	syl 17	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ≤ 𝐵)
37	34, 36	eqbrtrd 5122	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ≤ 𝐵)
38	26, 28, 29, 32, 37	letrd 11340	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ 𝐵)
39	3, 5, 10, 23, 38	elfzd 13520	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ (1...𝐵))
40	39	adantrr 727	. . . . 5 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ (𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵))) → (𝑏 − 𝑎) ∈ (1...𝐵))
41		rpre 13002	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42	41	ad3antrrr 740	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℝ)
43	42, 25	remulcld 11212	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℝ)
44	42, 24	remulcld 11212	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℝ)
45		simpr 488	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
46	25, 24, 45	ltled 11331	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ≤ 𝑏)
47		rpgt0 13006	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
48	47	ad3antrrr 740	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < 𝐴)
49		lemul2 12044	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝑎 ≤ 𝑏 ↔ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)))
50	25, 24, 42, 48, 49	syl112anc 1393	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑎 ≤ 𝑏 ↔ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)))
51	46, 50	mpbid 234	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ≤ (𝐴 · 𝑏))
52		flword2 13823	. . . . . . . 8 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ (𝐴 · 𝑏) ∈ ℝ ∧ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
53	43, 44, 51, 52	syl3anc 1390	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
54		uznn0sub 12874	. . . . . . 7 ⊢ ((⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))) → ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) ∈ ℕ₀)
55	53, 54	syl 17	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) ∈ ℕ₀)
56	55	adantrr 727	. . . . 5 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ (𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵))) → ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) ∈ ℕ₀)
57	42	recnd 11210	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℂ)
58	25	recnd 11210	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℂ)
59	57, 33, 58	subdid 11643	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · (𝑏 − 𝑎)) = ((𝐴 · 𝑏) − (𝐴 · 𝑎)))
60	59	oveq1d 7411	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))) = (((𝐴 · 𝑏) − (𝐴 · 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))))
61	44	recnd 11210	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℂ)
62	43	recnd 11210	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℂ)
63	44	flcld 13808	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℤ)
64	63	zcnd 12678	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℂ)
65	43	flcld 13808	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℤ)
66	65	zcnd 12678	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℂ)
67	61, 62, 64, 66	sub4d 11591	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (((𝐴 · 𝑏) − (𝐴 · 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))) = (((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))) − ((𝐴 · 𝑎) − (⌊‘(𝐴 · 𝑎)))))
68		modfrac 13894	. . . . . . . . . . . . . 14 ⊢ ((𝐴 · 𝑏) ∈ ℝ → ((𝐴 · 𝑏) mod 1) = ((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))))
69	44, 68	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) = ((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))))
70	69	eqcomd 2768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))) = ((𝐴 · 𝑏) mod 1))
71		modfrac 13894	. . . . . . . . . . . . . 14 ⊢ ((𝐴 · 𝑎) ∈ ℝ → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑎) − (⌊‘(𝐴 · 𝑎))))
72	43, 71	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) = ((𝐴 · 𝑎) − (⌊‘(𝐴 · 𝑎))))
73	72	eqcomd 2768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) − (⌊‘(𝐴 · 𝑎))) = ((𝐴 · 𝑎) mod 1))
74	70, 73	oveq12d 7414	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))) − ((𝐴 · 𝑎) − (⌊‘(𝐴 · 𝑎)))) = (((𝐴 · 𝑏) mod 1) − ((𝐴 · 𝑎) mod 1)))
75	60, 67, 74	3eqtrd 2801	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))) = (((𝐴 · 𝑏) mod 1) − ((𝐴 · 𝑎) mod 1)))
76	75	fveq2d 6871	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) = (abs‘(((𝐴 · 𝑏) mod 1) − ((𝐴 · 𝑎) mod 1))))
77		1rp 12997	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
78	77	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ∈ ℝ⁺)
79	44, 78	modcld 13885	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℝ)
80	79	recnd 11210	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℂ)
81	43, 78	modcld 13885	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
82	81	recnd 11210	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℂ)
83	80, 82	abssubd 15483	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (abs‘(((𝐴 · 𝑏) mod 1) − ((𝐴 · 𝑎) mod 1))) = (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))))
84	76, 83	eqtr2d 2798	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) = (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))))
85	84	breq1d 5110	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵) ↔ (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) < (1 / 𝐵)))
86	85	biimpd 231	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵) → (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) < (1 / 𝐵)))
87	86	impr 458	. . . . 5 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ (𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵))) → (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) < (1 / 𝐵))
88		oveq2 7404	. . . . . . . 8 ⊢ (𝑥 = (𝑏 − 𝑎) → (𝐴 · 𝑥) = (𝐴 · (𝑏 − 𝑎)))
89	88	fvoveq1d 7418	. . . . . . 7 ⊢ (𝑥 = (𝑏 − 𝑎) → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)))
90	89	breq1d 5110	. . . . . 6 ⊢ (𝑥 = (𝑏 − 𝑎) → ((abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵) ↔ (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) < (1 / 𝐵)))
91		oveq2 7404	. . . . . . . 8 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → ((𝐴 · (𝑏 − 𝑎)) − 𝑦) = ((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))))
92	91	fveq2d 6871	. . . . . . 7 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))))
93	92	breq1d 5110	. . . . . 6 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → ((abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) < (1 / 𝐵) ↔ (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) < (1 / 𝐵)))
94	90, 93	rspc2ev 3594	. . . . 5 ⊢ (((𝑏 − 𝑎) ∈ (1...𝐵) ∧ ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) ∈ ℕ₀ ∧ (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))) < (1 / 𝐵)) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵))
95	40, 56, 87, 94	syl3anc 1390	. . . 4 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ (𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵))) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵))
96	95	ex 416	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → ((𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵)) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵)))
97	96	rexlimdvva 3219	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (∃𝑎 ∈ (0...𝐵)∃𝑏 ∈ (0...𝐵)(𝑎 < 𝑏 ∧ (abs‘(((𝐴 · 𝑎) mod 1) − ((𝐴 · 𝑏) mod 1))) < (1 / 𝐵)) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵)))
98	1, 97	mpd 15	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ (1...𝐵)∃𝑦 ∈ ℕ₀ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1560 ∈ wcel 2142 ∃wrex 3086 class class class wbr 5100 ‘cfv 6521 (class class class)co 7396 ℝcr 11072 0cc0 11073 1c1 11074 · cmul 11078 < clt 11216 ≤ cle 11217 − cmin 11414 / cdiv 11844 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ℤ_≥cuz 12839 ℝ⁺crp 12993 ...cfz 13512 ⌊cfl 13800 mod cmo 13879 abscabs 15261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9388 df-inf 9389 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-n0 12482 df-xnn0 12555 df-z 12569 df-uz 12840 df-rp 12994 df-ico 13355 df-fz 13513 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263

This theorem is referenced by: irrapxlem4 43402

W3C validator