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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 15562 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) | |
| 2 | 1 | simplbi 497 | . 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
| 3 | cnex 11236 | . . . 4 ⊢ ℂ ∈ V | |
| 4 | reex 11246 | . . . 4 ⊢ ℝ ∈ V | |
| 5 | 3, 4 | elpm2 8914 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
| 6 | 5 | simplbi 497 | . 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 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑pm cpm 8867 ℂcc 11153 ℝcr 11154 +∞cpnf 11292 ≤ cle 11296 [,)cico 13389 abscabs 15273 𝑂(1)co1 15522 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 df-o1 15526 |
| This theorem is referenced by: o1res 15596 o1of2 15649 o1rlimmul 15655 o1mptrcl 15659 o1fsum 15849 o1cxp 27018 dchrisum0 27564 |
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