Theorem elo1 15562

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)

Assertion

Ref	Expression
elo1	⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))

Distinct variable group:   𝑥,𝑚,𝑦,𝐹

Proof of Theorem elo1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmeq 5914	. . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
2	1	ineq1d 4219	. . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3		fveq1 6905	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
4	3	fveq2d 6910	. . . . 5 ⊢ (𝑓 = 𝐹 → (abs‘(𝑓‘𝑦)) = (abs‘(𝐹‘𝑦)))
5	4	breq1d 5153	. . . 4 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑚))
6	2, 5	raleqbidv 3346	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
7	6	2rexbidv 3222	. 2 ⊢ (𝑓 = 𝐹 → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
8		df-o1 15526	. 2 ⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
9	7, 8	elrab2 3695	1 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   class class class wbr 5143  dom cdm 5685  ‘cfv 6561  (class class class)co 7431   ↑_pm cpm 8867  ℂcc 11153  ℝcr 11154  +∞cpnf 11292   ≤ cle 11296  [,)cico 13389  abscabs 15273  𝑂(1)co1 15522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-o1 15526

This theorem is referenced by:  elo12  15563  o1f  15565  o1dm  15566