irrapxlem4 - Mathbox for Stefan O'Rear

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Theorem irrapxlem4 43182

Description: Lemma for irrapx1 43185. Eliminate ranges, use positivity of the input to force positivity of the output by increasing 𝐵 as needed. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
irrapxlem4	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

Proof of Theorem irrapxlem4 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn 13481	. . . 4 ⊢ (𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) → 𝑎 ∈ ℕ)
2	1	ad3antlr 732	. . 3 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑎 ∈ ℕ)
3		nn0z 12524	. . . . 5 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℤ)
4	3	ad2antlr 728	. . . 4 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑏 ∈ ℤ)
5		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
6	5	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐴 ∈ ℝ⁺)
7	6	rpred 12961	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐴 ∈ ℝ)
8	2	nnred 12172	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑎 ∈ ℝ)
9	7, 8	remulcld 11174	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (𝐴 · 𝑎) ∈ ℝ)
10		nn0re 12422	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
11	10	ad2antlr 728	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑏 ∈ ℝ)
12	9, 11	resubcld 11577	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((𝐴 · 𝑎) − 𝑏) ∈ ℝ)
13	12	recnd 11172	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((𝐴 · 𝑎) − 𝑏) ∈ ℂ)
14	13	abscld 15374	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (abs‘((𝐴 · 𝑎) − 𝑏)) ∈ ℝ)
15	5	rpreccld 12971	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (1 / 𝐴) ∈ ℝ⁺)
16	15	rprege0d 12968	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)))
17		flge0nn0 13752	. . . . . . . . . . . 12 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ₀)
18		nn0p1nn 12452	. . . . . . . . . . . 12 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℕ₀ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
19	16, 17, 18	3syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
20	19	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
21		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
22	21	ad3antrrr 731	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐵 ∈ ℕ)
23	20, 22	ifcld 4528	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℕ)
24	23	nnrecred 12208	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ∈ ℝ)
25		0red 11147	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 ∈ ℝ)
26	9, 25	resubcld 11577	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((𝐴 · 𝑎) − 0) ∈ ℝ)
27		simpr 484	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)))
28	20	nnrecred 12208	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / ((⌊‘(1 / 𝐴)) + 1)) ∈ ℝ)
29	22	nnred 12172	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐵 ∈ ℝ)
30	6	rprecred 12972	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / 𝐴) ∈ ℝ)
31	30	flcld 13730	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (⌊‘(1 / 𝐴)) ∈ ℤ)
32	31	zred 12608	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (⌊‘(1 / 𝐴)) ∈ ℝ)
33		peano2re 11318	. . . . . . . . . . . . 13 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℝ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ)
34	32, 33	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ)
35		max2 13114	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → ((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
36	29, 34, 35	syl2anc 585	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
37	20	nngt0d 12206	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < ((⌊‘(1 / 𝐴)) + 1))
38	23	nnred 12172	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℝ)
39	23	nngt0d 12206	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
40		lerec 12037	. . . . . . . . . . . 12 ⊢ (((((⌊‘(1 / 𝐴)) + 1) ∈ ℝ ∧ 0 < ((⌊‘(1 / 𝐴)) + 1)) ∧ (if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℝ ∧ 0 < if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / ((⌊‘(1 / 𝐴)) + 1))))
41	34, 37, 38, 39, 40	syl22anc 839	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / ((⌊‘(1 / 𝐴)) + 1))))
42	36, 41	mpbid 232	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / ((⌊‘(1 / 𝐴)) + 1)))
43		fllep1 13733	. . . . . . . . . . . . 13 ⊢ ((1 / 𝐴) ∈ ℝ → (1 / 𝐴) ≤ ((⌊‘(1 / 𝐴)) + 1))
44	30, 43	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / 𝐴) ≤ ((⌊‘(1 / 𝐴)) + 1))
45	20	nncnd 12173	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((⌊‘(1 / 𝐴)) + 1) ∈ ℂ)
46	20	nnne0d 12207	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((⌊‘(1 / 𝐴)) + 1) ≠ 0)
47	45, 46	recrecd 11926	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / (1 / ((⌊‘(1 / 𝐴)) + 1))) = ((⌊‘(1 / 𝐴)) + 1))
48	44, 47	breqtrrd 5128	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / 𝐴) ≤ (1 / (1 / ((⌊‘(1 / 𝐴)) + 1))))
49	34, 37	recgt0d 12088	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < (1 / ((⌊‘(1 / 𝐴)) + 1)))
50	6	rpgt0d 12964	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < 𝐴)
51		lerec 12037	. . . . . . . . . . . 12 ⊢ ((((1 / ((⌊‘(1 / 𝐴)) + 1)) ∈ ℝ ∧ 0 < (1 / ((⌊‘(1 / 𝐴)) + 1))) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → ((1 / ((⌊‘(1 / 𝐴)) + 1)) ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / (1 / ((⌊‘(1 / 𝐴)) + 1)))))
52	28, 49, 7, 50, 51	syl22anc 839	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((1 / ((⌊‘(1 / 𝐴)) + 1)) ≤ 𝐴 ↔ (1 / 𝐴) ≤ (1 / (1 / ((⌊‘(1 / 𝐴)) + 1)))))
53	48, 52	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / ((⌊‘(1 / 𝐴)) + 1)) ≤ 𝐴)
54	24, 28, 7, 42, 53	letrd 11302	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ 𝐴)
55	7	recnd 11172	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐴 ∈ ℂ)
56	55	mulridd 11161	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (𝐴 · 1) = 𝐴)
57	2	nnge1d 12205	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 1 ≤ 𝑎)
58		1red 11145	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 1 ∈ ℝ)
59	58, 8, 6	lemul2d 13005	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 ≤ 𝑎 ↔ (𝐴 · 1) ≤ (𝐴 · 𝑎)))
60	57, 59	mpbid 232	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (𝐴 · 1) ≤ (𝐴 · 𝑎))
61	56, 60	eqbrtrrd 5124	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐴 ≤ (𝐴 · 𝑎))
62	9	recnd 11172	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (𝐴 · 𝑎) ∈ ℂ)
63	62	subid1d 11493	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((𝐴 · 𝑎) − 0) = (𝐴 · 𝑎))
64	61, 63	breqtrrd 5128	. . . . . . . . 9 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐴 ≤ ((𝐴 · 𝑎) − 0))
65	24, 7, 26, 54, 64	letrd 11302	. . . . . . . 8 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ ((𝐴 · 𝑎) − 0))
66	14, 24, 26, 27, 65	ltletrd 11305	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (abs‘((𝐴 · 𝑎) − 𝑏)) < ((𝐴 · 𝑎) − 0))
67	12, 26	absltd 15367	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((abs‘((𝐴 · 𝑎) − 𝑏)) < ((𝐴 · 𝑎) − 0) ↔ (-((𝐴 · 𝑎) − 0) < ((𝐴 · 𝑎) − 𝑏) ∧ ((𝐴 · 𝑎) − 𝑏) < ((𝐴 · 𝑎) − 0))))
68	66, 67	mpbid 232	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (-((𝐴 · 𝑎) − 0) < ((𝐴 · 𝑎) − 𝑏) ∧ ((𝐴 · 𝑎) − 𝑏) < ((𝐴 · 𝑎) − 0)))
69	68	simprd 495	. . . . 5 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ((𝐴 · 𝑎) − 𝑏) < ((𝐴 · 𝑎) − 0))
70	25, 11, 9	ltsub2d 11759	. . . . 5 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (0 < 𝑏 ↔ ((𝐴 · 𝑎) − 𝑏) < ((𝐴 · 𝑎) − 0)))
71	69, 70	mpbird 257	. . . 4 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < 𝑏)
72		elnnz 12510	. . . 4 ⊢ (𝑏 ∈ ℕ ↔ (𝑏 ∈ ℤ ∧ 0 < 𝑏))
73	4, 71, 72	sylanbrc 584	. . 3 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑏 ∈ ℕ)
74	22, 2	ifcld 4528	. . . . 5 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ∈ ℕ)
75	74	nnrecred 12208	. . . 4 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)) ∈ ℝ)
76		elfzle2 13456	. . . . . . 7 ⊢ (𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) → 𝑎 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
77	76	ad3antlr 732	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝑎 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
78		max1 13112	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
79	29, 34, 78	syl2anc 585	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
80		maxle 13118	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℝ) → (if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (𝑎 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∧ 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))))
81	8, 29, 38, 80	syl3anc 1374	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (𝑎 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∧ 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))))
82	77, 79, 81	mpbir2and 714	. . . . 5 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
83	29, 8	ifcld 4528	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ∈ ℝ)
84	22	nngt0d 12206	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < 𝐵)
85		max2 13114	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
86	8, 29, 85	syl2anc 585	. . . . . . 7 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 𝐵 ≤ if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
87	25, 29, 83, 84, 86	ltletrd 11305	. . . . . 6 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → 0 < if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
88		lerec 12037	. . . . . 6 ⊢ (((if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ∈ ℝ ∧ 0 < if(𝑎 ≤ 𝐵, 𝐵, 𝑎)) ∧ (if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℝ ∧ 0 < if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎))))
89	83, 87, 38, 39, 88	syl22anc 839	. . . . 5 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (if(𝑎 ≤ 𝐵, 𝐵, 𝑎) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ↔ (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎))))
90	82, 89	mpbid 232	. . . 4 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) ≤ (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)))
91	14, 24, 75, 27, 90	ltletrd 11305	. . 3 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)))
92		oveq2 7376	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝐴 · 𝑥) = (𝐴 · 𝑎))
93	92	fvoveq1d 7390	. . . . 5 ⊢ (𝑥 = 𝑎 → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑦)))
94		breq1 5103	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ≤ 𝐵 ↔ 𝑎 ≤ 𝐵))
95		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
96	94, 95	ifbieq2d 4508	. . . . . 6 ⊢ (𝑥 = 𝑎 → if(𝑥 ≤ 𝐵, 𝐵, 𝑥) = if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
97	96	oveq2d 7384	. . . . 5 ⊢ (𝑥 = 𝑎 → (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)) = (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)))
98	93, 97	breq12d 5113	. . . 4 ⊢ (𝑥 = 𝑎 → ((abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)) ↔ (abs‘((𝐴 · 𝑎) − 𝑦)) < (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎))))
99		oveq2 7376	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝐴 · 𝑎) − 𝑦) = ((𝐴 · 𝑎) − 𝑏))
100	99	fveq2d 6846	. . . . 5 ⊢ (𝑦 = 𝑏 → (abs‘((𝐴 · 𝑎) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑏)))
101	100	breq1d 5110	. . . 4 ⊢ (𝑦 = 𝑏 → ((abs‘((𝐴 · 𝑎) − 𝑦)) < (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)) ↔ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎))))
102	98, 101	rspc2ev 3591	. . 3 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)))
103	2, 73, 91, 102	syl3anc 1374	. 2 ⊢ (((((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) ∧ 𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) ∧ 𝑏 ∈ ℕ₀) ∧ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)))
104	19, 21	ifcld 4528	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℕ)
105		irrapxlem3 43181	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℕ) → ∃𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))∃𝑏 ∈ ℕ₀ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)))
106	5, 104, 105	syl2anc 585	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))∃𝑏 ∈ ℕ₀ (abs‘((𝐴 · 𝑎) − 𝑏)) < (1 / if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)))
107	103, 106	r19.29vva 3198	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ (abs‘((𝐴 · 𝑥) − 𝑦)) < (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2114 ∃wrex 3062 ifcif 4481 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 < clt 11178 ≤ cle 11179 − cmin 11376 -cneg 11377 / cdiv 11806 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ℝ⁺crp 12917 ...cfz 13435 ⌊cfl 13722 abscabs 15169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-rp 12918 df-ico 13279 df-fz 13436 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171

This theorem is referenced by: irrapxlem5 43183

W3C validator