Theorem elo1 14885

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 5774	. . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
2	1	ineq1d 4190	. . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞)))
3		fveq1 6671	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
4	3	fveq2d 6676	. . . . 5 ⊢ (𝑓 = 𝐹 → (abs‘(𝑓‘𝑦)) = (abs‘(𝐹‘𝑦)))
5	4	breq1d 5078	. . . 4 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑚))
6	2, 5	raleqbidv 3403	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))
7	6	2rexbidv 3302	. 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 14849	. 2 ⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
9	7, 8	elrab2 3685	1 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∩ cin 3937   class class class wbr 5068  dom cdm 5557  ‘cfv 6357  (class class class)co 7158   ↑_pm cpm 8409  ℂcc 10537  ℝcr 10538  +∞cpnf 10674   ≤ cle 10678  [,)cico 12743  abscabs 14595  𝑂(1)co1 14845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-dm 5567  df-iota 6316  df-fv 6365  df-o1 14849

This theorem is referenced by:  elo12  14886  o1f  14888  o1dm  14889