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Mirrors > Home > MPE Home > Th. List > lo1o12 | Structured version Visualization version GIF version |
Description: A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about ≤𝑂(1) to 𝑂(1).) (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1o12.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
lo1o12 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lo1o12.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) | |
2 | 1 | fmpttd 6635 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
3 | lo1o1 14641 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ≤𝑂(1))) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ≤𝑂(1))) |
5 | eqidd 2827 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
6 | absf 14455 | . . . . . 6 ⊢ abs:ℂ⟶ℝ | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝜑 → abs:ℂ⟶ℝ) |
8 | 7 | feqmptd 6497 | . . . 4 ⊢ (𝜑 → abs = (𝑦 ∈ ℂ ↦ (abs‘𝑦))) |
9 | fveq2 6434 | . . . 4 ⊢ (𝑦 = 𝐵 → (abs‘𝑦) = (abs‘𝐵)) | |
10 | 1, 5, 8, 9 | fmptco 6647 | . . 3 ⊢ (𝜑 → (abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) = (𝑥 ∈ 𝐴 ↦ (abs‘𝐵))) |
11 | 10 | eleq1d 2892 | . 2 ⊢ (𝜑 → ((abs ∘ (𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ≤𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1))) |
12 | 4, 11 | bitrd 271 | 1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2166 ↦ cmpt 4953 ∘ ccom 5347 ⟶wf 6120 ‘cfv 6124 ℂcc 10251 ℝcr 10252 abscabs 14352 𝑂(1)co1 14595 ≤𝑂(1)clo1 14596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-pm 8126 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-2 11415 df-3 11416 df-n0 11620 df-z 11706 df-uz 11970 df-rp 12114 df-ico 12470 df-seq 13097 df-exp 13156 df-cj 14217 df-re 14218 df-im 14219 df-sqrt 14353 df-abs 14354 df-o1 14599 df-lo1 14600 |
This theorem is referenced by: elo1mpt 14643 elo1mpt2 14644 elo1d 14645 o1bdd2 14650 o1bddrp 14651 o1eq 14679 o1le 14761 pntrlog2bndlem1 25680 |
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