irrapxlem4 - Mathbox for Stefan O'Rear

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Theorem irrapxlem4 43941

Description: Lemma for irrapx1 43944. 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 13694	. . . 4 ⊢ (𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) → 𝑎 ∈ ℕ)
2	1	ad3antlr 744	. . 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 12724	. . . . 5 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℤ)
4	3	ad2antlr 740	. . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐴 ∈ ℝ⁺)
6	5	ad3antrrr 743	. . . . . . . . . . . . 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 13171	. . . . . . . . . . . 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 12357	. . . . . . . . . . . 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 11346	. . . . . . . . . . 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 12622	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
11	10	ad2antlr 740	. . . . . . . . . . 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 11751	. . . . . . . . . 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 11344	. . . . . . . . 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 15615	. . . . . . . 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 13181	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (1 / 𝐴) ∈ ℝ⁺)
16	15	rprege0d 13178	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)))
17		flge0nn0 13967	. . . . . . . . . . . 12 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ₀)
18		nn0p1nn 12652	. . . . . . . . . . . 12 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℕ₀ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
19	16, 17, 18	3syl 19	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ((⌊‘(1 / 𝐴)) + 1) ∈ ℕ)
20	19	ad3antrrr 743	. . . . . . . . . 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 490	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → 𝐵 ∈ ℕ)
22	21	ad3antrrr 743	. . . . . . . . . 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 12396	. . . . . . . 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 11318	. . . . . . . . 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 11751	. . . . . . . 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 490	. . . . . . . 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 12396	. . . . . . . . . 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 12357	. . . . . . . . . . . 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 13182	. . . . . . . . . . . . . . 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 13945	. . . . . . . . . . . . . 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 12810	. . . . . . . . . . . . 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 11490	. . . . . . . . . . . . 13 ⊢ ((⌊‘(1 / 𝐴)) ∈ ℝ → ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ)
34	32, 33	syl 18	. . . . . . . . . . . 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 13324	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → ((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
36	29, 34, 35	syl2anc 596	. . . . . . . . . . 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 12394	. . . . . . . . . . . 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 12357	. . . . . . . . . . . 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 12394	. . . . . . . . . . . 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 12207	. . . . . . . . . . . 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 852	. . . . . . . . . . 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 235	. . . . . . . . . 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 13948	. . . . . . . . . . . . 13 ⊢ ((1 / 𝐴) ∈ ℝ → (1 / 𝐴) ≤ ((⌊‘(1 / 𝐴)) + 1))
44	30, 43	syl 18	. . . . . . . . . . . 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 12358	. . . . . . . . . . . . 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 12395	. . . . . . . . . . . . 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 12097	. . . . . . . . . . . 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 5132	. . . . . . . . . . 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 12258	. . . . . . . . . . . 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 13174	. . . . . . . . . . . 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 12207	. . . . . . . . . . . 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 852	. . . . . . . . . . 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 260	. . . . . . . . . 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 11474	. . . . . . . . 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 11344	. . . . . . . . . . . 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 11333	. . . . . . . . . . 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 12393	. . . . . . . . . . . 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 11316	. . . . . . . . . . . . 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 13215	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 5128	. . . . . . . . . 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 11344	. . . . . . . . . . 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 11665	. . . . . . . . . 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 5132	. . . . . . . . 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 11474	. . . . . . . 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 11477	. . . . . . 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 15608	. . . . . . 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 235	. . . . . 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 501	. . . . 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 11933	. . . . 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 260	. . . 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 12710	. . . 4 ⊢ (𝑏 ∈ ℕ ↔ (𝑏 ∈ ℤ ∧ 0 < 𝑏))
73	4, 71, 72	sylanbrc 595	. . 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 12396	. . . 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 13668	. . . . . . 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 744	. . . . . 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 13322	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
79	29, 34, 78	syl2anc 596	. . . . . 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 13328	. . . . . . 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 1398	. . . . . 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 726	. . . . 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 12394	. . . . . . 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 13324	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
86	8, 29, 85	syl2anc 596	. . . . . . 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 11477	. . . . . 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 12207	. . . . . 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 852	. . . . 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 235	. . . 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 11477	. . 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 7432	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝐴 · 𝑥) = (𝐴 · 𝑎))
93	92	fvoveq1d 7446	. . . . 5 ⊢ (𝑥 = 𝑎 → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑦)))
94		breq1 5105	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ≤ 𝐵 ↔ 𝑎 ≤ 𝐵))
95		id 23	. . . . . . 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 7440	. . . . 5 ⊢ (𝑥 = 𝑎 → (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)) = (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)))
98	93, 97	breq12d 5115	. . . 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 7432	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝐴 · 𝑎) − 𝑦) = ((𝐴 · 𝑎) − 𝑏))
100	99	fveq2d 6894	. . . . 5 ⊢ (𝑦 = 𝑏 → (abs‘((𝐴 · 𝑎) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑏)))
101	100	breq1d 5112	. . . 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 3588	. . 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 1398	. 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 43940	. . 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 596	. 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 3222	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∈ wcel 2144 ∃wrex 3086 ifcif 4481 class class class wbr 5102 ‘cfv 6543 (class class class)co 7424 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 ≤ cle 11351 − cmin 11548 -cneg 11549 / cdiv 11980 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 ℝ⁺crp 13127 ...cfz 13646 ⌊cfl 13937 abscabs 15410

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-oadd 8487 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-sup 9441 df-inf 9442 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-n0 12614 df-xnn0 12687 df-z 12701 df-uz 12973 df-rp 13128 df-ico 13489 df-fz 13647 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412

This theorem is used by: irrapxlem5 43942

W3C validator