irrapxlem3 - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > irrapxlem3		Structured version Visualization version GIF version

Theorem irrapxlem3 40562

Description: Lemma for irrapx1 40566. 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 40561	. 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 12280	. . . . . . . 8 ⊢ 1 ∈ ℤ
3	2	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ∈ ℤ)
4		simpllr 772	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℕ)
5	4	nnzd 12354	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℤ)
6		simplrr 774	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ (0...𝐵))
7	6	elfzelzd 13186	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℤ)
8		simplrl 773	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (0...𝐵))
9	8	elfzelzd 13186	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℤ)
10	7, 9	zsubcld 12360	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℤ)
11		1m1e0 11975	. . . . . . . . 9 ⊢ (1 − 1) = 0
12		elfzelz 13185	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
13	12	ad2antrl 724	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℤ)
14	13	zred 12355	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℝ)
15		elfzelz 13185	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (0...𝐵) → 𝑏 ∈ ℤ)
16	15	ad2antll 725	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℤ)
17	16	zred 12355	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℝ)
18	14, 17	posdifd 11492	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → (𝑎 < 𝑏 ↔ 0 < (𝑏 − 𝑎)))
19	18	biimpa 476	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < (𝑏 − 𝑎))
20	11, 19	eqbrtrid 5105	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (1 − 1) < (𝑏 − 𝑎))
21		zlem1lt 12302	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (𝑏 − 𝑎) ∈ ℤ) → (1 ≤ (𝑏 − 𝑎) ↔ (1 − 1) < (𝑏 − 𝑎)))
22	2, 10, 21	sylancr 586	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (1 ≤ (𝑏 − 𝑎) ↔ (1 − 1) < (𝑏 − 𝑎)))
23	20, 22	mpbird 256	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ≤ (𝑏 − 𝑎))
24	7	zred 12355	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℝ)
25	9	zred 12355	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℝ)
26	24, 25	resubcld 11333	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℝ)
27		0red 10909	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 ∈ ℝ)
28	24, 27	resubcld 11333	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ∈ ℝ)
29	4	nnred 11918	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℝ)
30		elfzle1 13188	. . . . . . . . . 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 11522	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ (𝑏 − 0))
33	24	recnd 10934	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℂ)
34	33	subid1d 11251	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) = 𝑏)
35		elfzle2 13189	. . . . . . . . . 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 5092	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ≤ 𝐵)
38	26, 28, 29, 32, 37	letrd 11062	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ 𝐵)
39	3, 5, 10, 23, 38	elfzd 13176	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ (1...𝐵))
40	39	adantrr 713	. . . . 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 12667	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42	41	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℝ)
43	42, 25	remulcld 10936	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℝ)
44	42, 24	remulcld 10936	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℝ)
45		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
46	25, 24, 45	ltled 11053	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ≤ 𝑏)
47		rpgt0 12671	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
48	47	ad3antrrr 726	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < 𝐴)
49		lemul2 11758	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝑎 ≤ 𝑏 ↔ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)))
50	25, 24, 42, 48, 49	syl112anc 1372	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑎 ≤ 𝑏 ↔ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)))
51	46, 50	mpbid 231	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ≤ (𝐴 · 𝑏))
52		flword2 13461	. . . . . . . 8 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ (𝐴 · 𝑏) ∈ ℝ ∧ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
53	43, 44, 51, 52	syl3anc 1369	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
54		uznn0sub 12546	. . . . . . 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 713	. . . . 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 10934	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℂ)
58	25	recnd 10934	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℂ)
59	57, 33, 58	subdid 11361	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · (𝑏 − 𝑎)) = ((𝐴 · 𝑏) − (𝐴 · 𝑎)))
60	59	oveq1d 7270	. . . . . . . . . . 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 10934	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℂ)
62	43	recnd 10934	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℂ)
63	44	flcld 13446	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℤ)
64	63	zcnd 12356	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℂ)
65	43	flcld 13446	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℤ)
66	65	zcnd 12356	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℂ)
67	61, 62, 64, 66	sub4d 11311	. . . . . . . . . . 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 13532	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))) = ((𝐴 · 𝑏) mod 1))
71		modfrac 13532	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . 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 7273	. . . . . . . . . . 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 2782	. . . . . . . . . 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 6760	. . . . . . . . 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 12663	. . . . . . . . . . . . 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 13523	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℝ)
80	79	recnd 10934	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℂ)
81	43, 78	modcld 13523	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
82	81	recnd 10934	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℂ)
83	80, 82	abssubd 15093	. . . . . . . . 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 2779	. . . . . . . 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 5080	. . . . . . 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 228	. . . . . 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 454	. . . . 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 7263	. . . . . . . 8 ⊢ (𝑥 = (𝑏 − 𝑎) → (𝐴 · 𝑥) = (𝐴 · (𝑏 − 𝑎)))
89	88	fvoveq1d 7277	. . . . . . 7 ⊢ (𝑥 = (𝑏 − 𝑎) → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)))
90	89	breq1d 5080	. . . . . 6 ⊢ (𝑥 = (𝑏 − 𝑎) → ((abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵) ↔ (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) < (1 / 𝐵)))
91		oveq2 7263	. . . . . . . 8 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → ((𝐴 · (𝑏 − 𝑎)) − 𝑦) = ((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))))
92	91	fveq2d 6760	. . . . . . 7 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))))
93	92	breq1d 5080	. . . . . 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 3564	. . . . 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 1369	. . . 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 412	. . 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 3222	. 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 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 0cc0 10802 1c1 10803 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 ⌊cfl 13438 mod cmo 13517 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-fz 13169 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: irrapxlem4 40563

W3C validator