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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigoimp | Structured version Visualization version GIF version |
Description: The defining property of a function of order G(x). (Contributed by AV, 18-May-2020.) |
Ref | Expression |
---|---|
elbigoimp | ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . 2 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → 𝐹 ∈ (Ο‘𝐺)) | |
2 | elbigofrcl 47938 | . . . . 5 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) | |
3 | reex 11249 | . . . . . 6 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 8903 | . . . . 5 ⊢ (𝐺 ∈ (ℝ ↑pm ℝ) ↔ (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
5 | 2, 4 | sylib 217 | . . . 4 ⊢ (𝐹 ∈ (Ο‘𝐺) → (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
6 | 5 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
7 | 3simpc 1147 | . . 3 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → (𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺)) | |
8 | elbigo2 47940 | . . 3 ⊢ (((𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ) ∧ (𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺)) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))) | |
9 | 6, 7, 8 | syl2anc 582 | . 2 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ∈ (Ο‘𝐺) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦))))) |
10 | 1, 9 | mpbid 231 | 1 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝑚 · (𝐺‘𝑦)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 ⊆ wss 3947 class class class wbr 5153 dom cdm 5682 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ↑pm cpm 8856 ℝcr 11157 · cmul 11163 ≤ cle 11299 Οcbigo 47935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-ico 13384 df-bigo 47936 |
This theorem is referenced by: (None) |
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