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Mirrors > Home > MPE Home > Th. List > elo1 | Structured version Visualization version GIF version |
Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
elo1 | ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 5858 | . . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
2 | 1 | ineq1d 4170 | . . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞))) |
3 | fveq1 6839 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
4 | 3 | fveq2d 6844 | . . . . 5 ⊢ (𝑓 = 𝐹 → (abs‘(𝑓‘𝑦)) = (abs‘(𝐹‘𝑦))) |
5 | 4 | breq1d 5114 | . . . 4 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
6 | 2, 5 | raleqbidv 3318 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
7 | 6 | 2rexbidv 3212 | . 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 15332 | . 2 ⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚} | |
9 | 7, 8 | elrab2 3647 | 1 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ∃wrex 3072 ∩ cin 3908 class class class wbr 5104 dom cdm 5632 ‘cfv 6494 (class class class)co 7352 ↑pm cpm 8725 ℂcc 11008 ℝcr 11009 +∞cpnf 11145 ≤ cle 11149 [,)cico 13221 abscabs 15079 𝑂(1)co1 15328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-dm 5642 df-iota 6446 df-fv 6502 df-o1 15332 |
This theorem is referenced by: elo12 15369 o1f 15371 o1dm 15372 |
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