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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 14883 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10618 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10628 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8438 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simplbi 500 | . 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 2114 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑pm cpm 8407 ℂcc 10535 ℝcr 10536 +∞cpnf 10672 ≤ cle 10676 [,)cico 12741 abscabs 14593 𝑂(1)co1 14843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-o1 14847 |
This theorem is referenced by: o1res 14917 o1of2 14969 o1rlimmul 14975 o1mptrcl 14979 o1fsum 15168 o1cxp 25552 dchrisum0 26096 |
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