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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigofrcl | Structured version Visualization version GIF version |
Description: Reverse closure of the "big-O" function. (Contributed by AV, 16-May-2020.) |
Ref | Expression |
---|---|
elbigofrcl | ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6695 | . 2 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ dom Ο) | |
2 | df-bigo 44678 | . . . 4 ⊢ Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) | |
3 | 2 | dmeqi 5766 | . . 3 ⊢ dom Ο = dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) |
4 | dmmptg 6089 | . . . 4 ⊢ (∀𝑔 ∈ (ℝ ↑pm ℝ){𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V → dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) = (ℝ ↑pm ℝ)) | |
5 | ovex 7182 | . . . . . 6 ⊢ (ℝ ↑pm ℝ) ∈ V | |
6 | 5 | rabex 5228 | . . . . 5 ⊢ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑔 ∈ (ℝ ↑pm ℝ) → {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))} ∈ V) |
8 | 4, 7 | mprg 3151 | . . 3 ⊢ dom (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓‘𝑦) ≤ (𝑚 · (𝑔‘𝑦))}) = (ℝ ↑pm ℝ) |
9 | 3, 8 | eqtri 2843 | . 2 ⊢ dom Ο = (ℝ ↑pm ℝ) |
10 | 1, 9 | eleqtrdi 2922 | 1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 {crab 3141 Vcvv 3491 ∩ cin 3928 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5548 ‘cfv 6348 (class class class)co 7149 ↑pm cpm 8400 ℝcr 10529 · cmul 10535 +∞cpnf 10665 ≤ cle 10669 [,)cico 12734 Οcbigo 44677 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7152 df-bigo 44678 |
This theorem is referenced by: elbigo 44681 elbigoimp 44686 |
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