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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 1135 | . 2 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → 𝐹 ∈ (Ο‘𝐺)) | |
2 | elbigofrcl 48400 | . . . . 5 ⊢ (𝐹 ∈ (Ο‘𝐺) → 𝐺 ∈ (ℝ ↑pm ℝ)) | |
3 | reex 11244 | . . . . . 6 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 8913 | . . . . 5 ⊢ (𝐺 ∈ (ℝ ↑pm ℝ) ↔ (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
5 | 2, 4 | sylib 218 | . . . 4 ⊢ (𝐹 ∈ (Ο‘𝐺) → (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
6 | 5 | 3ad2ant1 1132 | . . 3 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → (𝐺:dom 𝐺⟶ℝ ∧ dom 𝐺 ⊆ ℝ)) |
7 | 3simpc 1149 | . . 3 ⊢ ((𝐹 ∈ (Ο‘𝐺) ∧ 𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺) → (𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ dom 𝐺)) | |
8 | elbigo2 48402 | . . 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 584 | . 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 232 | 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 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑pm cpm 8866 ℝcr 11152 · cmul 11158 ≤ cle 11294 Οcbigo 48397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ico 13390 df-bigo 48398 |
This theorem is referenced by: (None) |
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