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Mirrors > Home > MPE Home > Th. List > o1le | Structured version Visualization version GIF version |
Description: Transfer eventual boundedness from a larger function to a smaller function. (Contributed by Mario Carneiro, 25-Sep-2014.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
o1le.1 | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
o1le.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) |
o1le.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
o1le.4 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
o1le.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → (abs‘𝐶) ≤ (abs‘𝐵)) |
Ref | Expression |
---|---|
o1le | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1le.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
2 | o1le.2 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1)) | |
3 | o1le.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
4 | 3, 2 | o1mptrcl 14201 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
5 | 4 | lo1o12 14112 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1))) |
6 | 2, 5 | mpbid 221 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐵)) ∈ ≤𝑂(1)) |
7 | fvex 6113 | . . . 4 ⊢ (abs‘𝐵) ∈ V | |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐵) ∈ V) |
9 | o1le.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) | |
10 | 9 | abscld 14023 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (abs‘𝐶) ∈ ℝ) |
11 | o1le.5 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑀 ≤ 𝑥)) → (abs‘𝐶) ≤ (abs‘𝐵)) | |
12 | 1, 6, 8, 10, 11 | lo1le 14230 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (abs‘𝐶)) ∈ ≤𝑂(1)) |
13 | 9 | lo1o12 14112 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1) ↔ (𝑥 ∈ 𝐴 ↦ (abs‘𝐶)) ∈ ≤𝑂(1))) |
14 | 12, 13 | mpbird 246 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 ℂcc 9813 ℝcr 9814 ≤ cle 9954 abscabs 13822 𝑂(1)co1 14065 ≤𝑂(1)clo1 14066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-o1 14069 df-lo1 14070 |
This theorem is referenced by: vmadivsum 24971 dchrvmasumlem2 24987 dchrvmasumlem3 24988 dchrvmasumiflem2 24991 dchrisum0fno1 25000 dchrisum0re 25002 dchrisum0lem1 25005 dchrisum0lem3 25008 mudivsum 25019 mulog2sumlem2 25024 2vmadivsumlem 25029 selberglem2 25035 selberg2lem 25039 selberg3lem1 25046 selberg4lem1 25049 pntrsumo1 25054 pntrlog2bndlem2 25067 pntrlog2bndlem3 25068 |
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