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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 5900 | . . . . 5 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
2 | 1 | ineq1d 4207 | . . . 4 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∩ (𝑥[,)+∞)) = (dom 𝐹 ∩ (𝑥[,)+∞))) |
3 | fveq1 6890 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
4 | 3 | fveq2d 6895 | . . . . 5 ⊢ (𝑓 = 𝐹 → (abs‘(𝑓‘𝑦)) = (abs‘(𝐹‘𝑦))) |
5 | 4 | breq1d 5152 | . . . 4 ⊢ (𝑓 = 𝐹 → ((abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
6 | 2, 5 | raleqbidv 3338 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚 ↔ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
7 | 6 | 2rexbidv 3215 | . 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 15460 | . 2 ⊢ 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚} | |
9 | 7, 8 | elrab2 3684 | 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 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 ∩ cin 3944 class class class wbr 5142 dom cdm 5672 ‘cfv 6542 (class class class)co 7414 ↑pm cpm 8839 ℂcc 11130 ℝcr 11131 +∞cpnf 11269 ≤ cle 11273 [,)cico 13352 abscabs 15207 𝑂(1)co1 15456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-dm 5682 df-iota 6494 df-fv 6550 df-o1 15460 |
This theorem is referenced by: elo12 15497 o1f 15499 o1dm 15500 |
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