irrapxlem3 - Mathbox for Stefan O'Rear

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Theorem irrapxlem3 42308

Description: Lemma for irrapx1 42312. 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 42307	. 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 12620	. . . . . . . 8 ⊢ 1 ∈ ℤ
3	2	a1i 11	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 1 ∈ ℤ)
4		simpllr 774	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℕ)
5	4	nnzd 12613	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℤ)
6		simplrr 776	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ (0...𝐵))
7	6	elfzelzd 13532	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℤ)
8		simplrl 775	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ (0...𝐵))
9	8	elfzelzd 13532	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℤ)
10	7, 9	zsubcld 12699	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℤ)
11		1m1e0 12312	. . . . . . . . 9 ⊢ (1 − 1) = 0
12		elfzelz 13531	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ (0...𝐵) → 𝑎 ∈ ℤ)
13	12	ad2antrl 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℤ)
14	13	zred 12694	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑎 ∈ ℝ)
15		elfzelz 13531	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (0...𝐵) → 𝑏 ∈ ℤ)
16	15	ad2antll 727	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℤ)
17	16	zred 12694	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → 𝑏 ∈ ℝ)
18	14, 17	posdifd 11829	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) → (𝑎 < 𝑏 ↔ 0 < (𝑏 − 𝑎)))
19	18	biimpa 475	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < (𝑏 − 𝑎))
20	11, 19	eqbrtrid 5178	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (1 − 1) < (𝑏 − 𝑎))
21		zlem1lt 12642	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ (𝑏 − 𝑎) ∈ ℤ) → (1 ≤ (𝑏 − 𝑎) ↔ (1 − 1) < (𝑏 − 𝑎)))
22	2, 10, 21	sylancr 585	. . . . . . . 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 12694	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℝ)
25	9	zred 12694	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℝ)
26	24, 25	resubcld 11670	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ ℝ)
27		0red 11245	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 ∈ ℝ)
28	24, 27	resubcld 11670	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ∈ ℝ)
29	4	nnred 12255	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐵 ∈ ℝ)
30		elfzle1 13534	. . . . . . . . . 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 11859	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ (𝑏 − 0))
33	24	recnd 11270	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑏 ∈ ℂ)
34	33	subid1d 11588	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) = 𝑏)
35		elfzle2 13535	. . . . . . . . . 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 5165	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 0) ≤ 𝐵)
38	26, 28, 29, 32, 37	letrd 11399	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ≤ 𝐵)
39	3, 5, 10, 23, 38	elfzd 13522	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝑏 − 𝑎) ∈ (1...𝐵))
40	39	adantrr 715	. . . . 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 13012	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42	41	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℝ)
43	42, 25	remulcld 11272	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℝ)
44	42, 24	remulcld 11272	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℝ)
45		simpr 483	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 < 𝑏)
46	25, 24, 45	ltled 11390	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ≤ 𝑏)
47		rpgt0 13016	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
48	47	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 0 < 𝐴)
49		lemul2 12095	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝑎 ≤ 𝑏 ↔ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)))
50	25, 24, 42, 48, 49	syl112anc 1371	. . . . . . . . 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 13808	. . . . . . . 8 ⊢ (((𝐴 · 𝑎) ∈ ℝ ∧ (𝐴 · 𝑏) ∈ ℝ ∧ (𝐴 · 𝑎) ≤ (𝐴 · 𝑏)) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
53	43, 44, 51, 52	syl3anc 1368	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ (ℤ_≥‘(⌊‘(𝐴 · 𝑎))))
54		uznn0sub 12889	. . . . . . 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 715	. . . . 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 11270	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝐴 ∈ ℂ)
58	25	recnd 11270	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → 𝑎 ∈ ℂ)
59	57, 33, 58	subdid 11698	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · (𝑏 − 𝑎)) = ((𝐴 · 𝑏) − (𝐴 · 𝑎)))
60	59	oveq1d 7430	. . . . . . . . . . 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 11270	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑏) ∈ ℂ)
62	43	recnd 11270	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (𝐴 · 𝑎) ∈ ℂ)
63	44	flcld 13793	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℤ)
64	63	zcnd 12695	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑏)) ∈ ℂ)
65	43	flcld 13793	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℤ)
66	65	zcnd 12695	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → (⌊‘(𝐴 · 𝑎)) ∈ ℂ)
67	61, 62, 64, 66	sub4d 11648	. . . . . . . . . . 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 13879	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) − (⌊‘(𝐴 · 𝑏))) = ((𝐴 · 𝑏) mod 1))
71		modfrac 13879	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . 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 7433	. . . . . . . . . . 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 2769	. . . . . . . . . 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 6895	. . . . . . . . 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 13008	. . . . . . . . . . . . 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 13870	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℝ)
80	79	recnd 11270	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑏) mod 1) ∈ ℂ)
81	43, 78	modcld 13870	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℝ)
82	81	recnd 11270	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ (𝑎 ∈ (0...𝐵) ∧ 𝑏 ∈ (0...𝐵))) ∧ 𝑎 < 𝑏) → ((𝐴 · 𝑎) mod 1) ∈ ℂ)
83	80, 82	abssubd 15430	. . . . . . . . 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 2766	. . . . . . . 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 5153	. . . . . . 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 453	. . . . 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 7423	. . . . . . . 8 ⊢ (𝑥 = (𝑏 − 𝑎) → (𝐴 · 𝑥) = (𝐴 · (𝑏 − 𝑎)))
89	88	fvoveq1d 7437	. . . . . . 7 ⊢ (𝑥 = (𝑏 − 𝑎) → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)))
90	89	breq1d 5153	. . . . . 6 ⊢ (𝑥 = (𝑏 − 𝑎) → ((abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / 𝐵) ↔ (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) < (1 / 𝐵)))
91		oveq2 7423	. . . . . . . 8 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → ((𝐴 · (𝑏 − 𝑎)) − 𝑦) = ((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎)))))
92	91	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))) → (abs‘((𝐴 · (𝑏 − 𝑎)) − 𝑦)) = (abs‘((𝐴 · (𝑏 − 𝑎)) − ((⌊‘(𝐴 · 𝑏)) − (⌊‘(𝐴 · 𝑎))))))
93	92	breq1d 5153	. . . . . 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 3615	. . . . 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 1368	. . . 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 411	. . 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 3202	. 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 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℝcr 11135 0cc0 11136 1c1 11137 · cmul 11141 < clt 11276 ≤ cle 11277 − cmin 11472 / cdiv 11899 ℕcn 12240 ℕ₀cn0 12500 ℤcz 12586 ℤ_≥cuz 12850 ℝ⁺crp 13004 ...cfz 13514 ⌊cfl 13785 mod cmo 13864 abscabs 15211

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-2 12303 df-3 12304 df-n0 12501 df-xnn0 12573 df-z 12587 df-uz 12851 df-rp 13005 df-ico 13360 df-fz 13515 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213

This theorem is referenced by: irrapxlem4 42309

W3C validator