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Mirrors > Home > MPE Home > Th. List > lo1o1 | Structured version Visualization version GIF version |
Description: A function is eventually bounded iff its absolute value is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1o1 | ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1dm 14261 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ) | |
2 | fdm 6051 | . . . 4 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴) | |
3 | 2 | sseq1d 3632 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → (dom 𝐹 ⊆ ℝ ↔ 𝐴 ⊆ ℝ)) |
4 | 1, 3 | syl5ib 234 | . 2 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) → 𝐴 ⊆ ℝ)) |
5 | lo1dm 14250 | . . 3 ⊢ ((abs ∘ 𝐹) ∈ ≤𝑂(1) → dom (abs ∘ 𝐹) ⊆ ℝ) | |
6 | absf 14077 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
7 | fco 6058 | . . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ 𝐹:𝐴⟶ℂ) → (abs ∘ 𝐹):𝐴⟶ℝ) | |
8 | 6, 7 | mpan 706 | . . . . 5 ⊢ (𝐹:𝐴⟶ℂ → (abs ∘ 𝐹):𝐴⟶ℝ) |
9 | fdm 6051 | . . . . 5 ⊢ ((abs ∘ 𝐹):𝐴⟶ℝ → dom (abs ∘ 𝐹) = 𝐴) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶ℂ → dom (abs ∘ 𝐹) = 𝐴) |
11 | 10 | sseq1d 3632 | . . 3 ⊢ (𝐹:𝐴⟶ℂ → (dom (abs ∘ 𝐹) ⊆ ℝ ↔ 𝐴 ⊆ ℝ)) |
12 | 5, 11 | syl5ib 234 | . 2 ⊢ (𝐹:𝐴⟶ℂ → ((abs ∘ 𝐹) ∈ ≤𝑂(1) → 𝐴 ⊆ ℝ)) |
13 | fvco3 6275 | . . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝑦 ∈ 𝐴) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦))) | |
14 | 13 | adantlr 751 | . . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ 𝐴) → ((abs ∘ 𝐹)‘𝑦) = (abs‘(𝐹‘𝑦))) |
15 | 14 | breq1d 4663 | . . . . . . 7 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ 𝐴) → (((abs ∘ 𝐹)‘𝑦) ≤ 𝑚 ↔ (abs‘(𝐹‘𝑦)) ≤ 𝑚)) |
16 | 15 | imbi2d 330 | . . . . . 6 ⊢ (((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ 𝐴) → ((𝑥 ≤ 𝑦 → ((abs ∘ 𝐹)‘𝑦) ≤ 𝑚) ↔ (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚))) |
17 | 16 | ralbidva 2985 | . . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((abs ∘ 𝐹)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚))) |
18 | 17 | 2rexbidv 3057 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((abs ∘ 𝐹)‘𝑦) ≤ 𝑚) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚))) |
19 | ello12 14247 | . . . . 5 ⊢ (((abs ∘ 𝐹):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((abs ∘ 𝐹) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((abs ∘ 𝐹)‘𝑦) ≤ 𝑚))) | |
20 | 8, 19 | sylan 488 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → ((abs ∘ 𝐹) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → ((abs ∘ 𝐹)‘𝑦) ≤ 𝑚))) |
21 | elo12 14258 | . . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (abs‘(𝐹‘𝑦)) ≤ 𝑚))) | |
22 | 18, 20, 21 | 3bitr4rd 301 | . . 3 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1))) |
23 | 22 | ex 450 | . 2 ⊢ (𝐹:𝐴⟶ℂ → (𝐴 ⊆ ℝ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1)))) |
24 | 4, 12, 23 | pm5.21ndd 369 | 1 ⊢ (𝐹:𝐴⟶ℂ → (𝐹 ∈ 𝑂(1) ↔ (abs ∘ 𝐹) ∈ ≤𝑂(1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 ℂcc 9934 ℝcr 9935 ≤ cle 10075 abscabs 13974 𝑂(1)co1 14217 ≤𝑂(1)clo1 14218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-o1 14221 df-lo1 14222 |
This theorem is referenced by: lo1o12 14264 o1res 14291 |
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