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Mirrors > Home > MPE Home > Th. List > elo12r | Structured version Visualization version GIF version |
Description: Sufficient condition for elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
elo12r | ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5151 | . . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥)) | |
2 | 1 | imbi1d 341 | . . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))) |
3 | 2 | ralbidv 3176 | . . . . 5 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))) |
4 | breq2 5152 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ((abs‘(𝐹‘𝑥)) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑥)) ≤ 𝑀)) | |
5 | 4 | imbi2d 340 | . . . . . 6 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀))) |
6 | 5 | ralbidv 3176 | . . . . 5 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀))) |
7 | 3, 6 | rspc2ev 3635 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)) |
8 | 7 | 3expa 1117 | . . 3 ⊢ (((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)) |
9 | 8 | 3adant1 1129 | . 2 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚)) |
10 | elo12 15560 | . . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))) | |
11 | 10 | 3ad2ant1 1132 | . 2 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑚))) |
12 | 9, 11 | mpbird 257 | 1 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ) ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → (abs‘(𝐹‘𝑥)) ≤ 𝑀)) → 𝐹 ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 ℂcc 11151 ℝcr 11152 ≤ cle 11294 abscabs 15270 𝑂(1)co1 15519 |
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-o1 15523 |
This theorem is referenced by: o1resb 15599 o1of2 15646 o1cxp 27033 |
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