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Mirrors > Home > MPE Home > Th. List > o1f | Structured version Visualization version GIF version |
Description: An eventually bounded function is a function. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
o1f | ⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elo1 14966 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) | |
2 | 1 | simplbi 501 | . 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10689 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10699 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8477 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simplbi 501 | . 2 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℂ) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹:dom 𝐹⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∀wral 3053 ∃wrex 3054 ∩ cin 3840 ⊆ wss 3841 class class class wbr 5027 dom cdm 5519 ⟶wf 6329 ‘cfv 6333 (class class class)co 7164 ↑pm cpm 8431 ℂcc 10606 ℝcr 10607 +∞cpnf 10743 ≤ cle 10747 [,)cico 12816 abscabs 14676 𝑂(1)co1 14926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-pm 8433 df-o1 14930 |
This theorem is referenced by: o1res 15000 o1of2 15053 o1rlimmul 15059 o1mptrcl 15063 o1fsum 15254 o1cxp 25704 dchrisum0 26248 |
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