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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigof | Structured version Visualization version GIF version |
Description: A function of order G(x) is a function. (Contributed by AV, 18-May-2020.) |
Ref | Expression |
---|---|
elbigof | ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbigo 47139 | . 2 ⊢ (𝐹 ∈ (Ο‘𝐺) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) | |
2 | reex 11197 | . . . . 5 ⊢ ℝ ∈ V | |
3 | 2, 2 | elpm2 8864 | . . . 4 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
4 | 3 | simplbi 499 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℝ) |
5 | 4 | 3ad2ant1 1134 | . 2 ⊢ ((𝐹 ∈ (ℝ ↑pm ℝ) ∧ 𝐺 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))) → 𝐹:dom 𝐹⟶ℝ) |
6 | 1, 5 | sylbi 216 | 1 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐹:dom 𝐹⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ↑pm cpm 8817 ℝcr 11105 · cmul 11111 +∞cpnf 11241 ≤ cle 11245 [,)cico 13322 Οcbigo 47135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-pm 8819 df-bigo 47136 |
This theorem is referenced by: (None) |
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