irrapxlem4 - Mathbox for Stefan O'Rear

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Theorem irrapxlem4 42781

Description: Lemma for irrapx1 42784. 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 13613	. . . 4 ⊢ (𝑎 ∈ (1...if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵)) → 𝑎 ∈ ℕ)
2	1	ad3antlr 730	. . 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 12664	. . . . 5 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℤ)
4	3	ad2antlr 726	. . . 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 729	. . . . . . . . . . . . 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 13099	. . . . . . . . . . . 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 12308	. . . . . . . . . . . 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 11320	. . . . . . . . . . 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 12562	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
11	10	ad2antlr 726	. . . . . . . . . . 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 11718	. . . . . . . . . 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 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))) → ((𝐴 · 𝑎) − 𝑏) ∈ ℂ)
14	13	abscld 15485	. . . . . . . 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 13109	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → (1 / 𝐴) ∈ ℝ⁺)
16	15	rprege0d 13106	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → ((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)))
17		flge0nn0 13871	. . . . . . . . . . . 12 ⊢ (((1 / 𝐴) ∈ ℝ ∧ 0 ≤ (1 / 𝐴)) → (⌊‘(1 / 𝐴)) ∈ ℕ₀)
18		nn0p1nn 12592	. . . . . . . . . . . 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 729	. . . . . . . . . 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 729	. . . . . . . . . 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 4594	. . . . . . . . 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 12344	. . . . . . . 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 11293	. . . . . . . . 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 11718	. . . . . . . 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 12344	. . . . . . . . . 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 12308	. . . . . . . . . . . 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 13110	. . . . . . . . . . . . . . 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 13849	. . . . . . . . . . . . . 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 12747	. . . . . . . . . . . . 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 11463	. . . . . . . . . . . . 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 13249	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → ((⌊‘(1 / 𝐴)) + 1) ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
36	29, 34, 35	syl2anc 583	. . . . . . . . . . 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 12342	. . . . . . . . . . . 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 12308	. . . . . . . . . . . 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 12342	. . . . . . . . . . . 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 12178	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 13852	. . . . . . . . . . . . 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 12309	. . . . . . . . . . . . 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 12343	. . . . . . . . . . . . 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 12067	. . . . . . . . . . . 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 5194	. . . . . . . . . . 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 12229	. . . . . . . . . . . 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 13102	. . . . . . . . . . . 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 12178	. . . . . . . . . . . 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 838	. . . . . . . . . . 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 11447	. . . . . . . . 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 11318	. . . . . . . . . . . 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 11307	. . . . . . . . . . 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 12341	. . . . . . . . . . . 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 11291	. . . . . . . . . . . . 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 13143	. . . . . . . . . . . 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 5190	. . . . . . . . . 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 11318	. . . . . . . . . . 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 11636	. . . . . . . . . 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 5194	. . . . . . . . 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 11447	. . . . . . . 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 11450	. . . . . . 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 15478	. . . . . . 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 11900	. . . . 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 12649	. . . 4 ⊢ (𝑏 ∈ ℕ ↔ (𝑏 ∈ ℤ ∧ 0 < 𝑏))
73	4, 71, 72	sylanbrc 582	. . 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 4594	. . . . 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 12344	. . . 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 13588	. . . . . . 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 730	. . . . . 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 13247	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ ((⌊‘(1 / 𝐴)) + 1) ∈ ℝ) → 𝐵 ≤ if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵))
79	29, 34, 78	syl2anc 583	. . . . . 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 13253	. . . . . . 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 1371	. . . . . 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 712	. . . . 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 4594	. . . . . 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 12342	. . . . . . 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 13249	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
86	8, 29, 85	syl2anc 583	. . . . . . 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 11450	. . . . . 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 12178	. . . . . 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 838	. . . . 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 11450	. . 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 7456	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝐴 · 𝑥) = (𝐴 · 𝑎))
93	92	fvoveq1d 7470	. . . . 5 ⊢ (𝑥 = 𝑎 → (abs‘((𝐴 · 𝑥) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑦)))
94		breq1 5169	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ≤ 𝐵 ↔ 𝑎 ≤ 𝐵))
95		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
96	94, 95	ifbieq2d 4574	. . . . . 6 ⊢ (𝑥 = 𝑎 → if(𝑥 ≤ 𝐵, 𝐵, 𝑥) = if(𝑎 ≤ 𝐵, 𝐵, 𝑎))
97	96	oveq2d 7464	. . . . 5 ⊢ (𝑥 = 𝑎 → (1 / if(𝑥 ≤ 𝐵, 𝐵, 𝑥)) = (1 / if(𝑎 ≤ 𝐵, 𝐵, 𝑎)))
98	93, 97	breq12d 5179	. . . 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 7456	. . . . . 6 ⊢ (𝑦 = 𝑏 → ((𝐴 · 𝑎) − 𝑦) = ((𝐴 · 𝑎) − 𝑏))
100	99	fveq2d 6924	. . . . 5 ⊢ (𝑦 = 𝑏 → (abs‘((𝐴 · 𝑎) − 𝑦)) = (abs‘((𝐴 · 𝑎) − 𝑏)))
101	100	breq1d 5176	. . . 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 3648	. . 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 1371	. 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 4594	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐵 ∈ ℕ) → if(𝐵 ≤ ((⌊‘(1 / 𝐴)) + 1), ((⌊‘(1 / 𝐴)) + 1), 𝐵) ∈ ℕ)
105		irrapxlem3 42780	. . 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 583	. 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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∃wrex 3076 ifcif 4548 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ℝ⁺crp 13057 ...cfz 13567 ⌊cfl 13841 abscabs 15283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-fz 13568 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285

This theorem is referenced by: irrapxlem5 42782

W3C validator