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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 14875 | . . 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 10607 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10617 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8421 | . . 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 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℂcc 10524 ℝcr 10525 +∞cpnf 10661 ≤ cle 10665 [,)cico 12728 abscabs 14585 𝑂(1)co1 14835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-o1 14839 |
This theorem is referenced by: o1res 14909 o1of2 14961 o1rlimmul 14967 o1mptrcl 14971 o1fsum 15160 o1cxp 25560 dchrisum0 26104 |
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