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Mirrors > Home > MPE Home > Th. List > chpo1ub | Structured version Visualization version GIF version |
Description: The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.) |
Ref | Expression |
---|---|
chpo1ub | ⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11712 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
2 | elicopnf 12834 | . . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))) | |
3 | 1, 2 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)) |
4 | chtrpcl 25752 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ+) | |
5 | 3, 4 | sylbi 219 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ+) |
6 | 5 | rpcnne0d 12441 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0)) |
7 | 3 | simplbi 500 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ) |
8 | 0red 10644 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ) | |
9 | 1 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ) |
10 | 2pos 11741 | . . . . . . . . . . . 12 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2) |
12 | 3 | simprbi 499 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥) |
13 | 8, 9, 7, 11, 12 | ltletrd 10800 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥) |
14 | 7, 13 | elrpd 12429 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ+) |
15 | 14 | rpcnne0d 12441 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) |
16 | rpre 12398 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
17 | chpcl 25701 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ) | |
18 | 16, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ) |
19 | 18 | recnd 10669 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ) |
20 | 14, 19 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ) |
21 | dmdcan 11350 | . . . . . . . 8 ⊢ ((((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (ψ‘𝑥) ∈ ℂ) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥)) | |
22 | 6, 15, 20, 21 | syl3anc 1367 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥)) |
23 | 22 | adantl 484 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥)) |
24 | 23 | mpteq2dva 5161 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥))) |
25 | ovexd 7191 | . . . . . 6 ⊢ (⊤ → (2[,)+∞) ∈ V) | |
26 | ovexd 7191 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((θ‘𝑥) / 𝑥) ∈ V) | |
27 | ovexd 7191 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ V) | |
28 | eqidd 2822 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))) | |
29 | eqidd 2822 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) | |
30 | 25, 26, 27, 28, 29 | offval2 7426 | . . . . 5 ⊢ (⊤ → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))))) |
31 | 14 | ssriv 3971 | . . . . . 6 ⊢ (2[,)+∞) ⊆ ℝ+ |
32 | resmpt 5905 | . . . . . 6 ⊢ ((2[,)+∞) ⊆ ℝ+ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥))) | |
33 | 31, 32 | mp1i 13 | . . . . 5 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥))) |
34 | 24, 30, 33 | 3eqtr4rd 2867 | . . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) = ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))))) |
35 | 31 | a1i 11 | . . . . . 6 ⊢ (⊤ → (2[,)+∞) ⊆ ℝ+) |
36 | chto1ub 26052 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1) | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
38 | 35, 37 | o1res2 14920 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
39 | chpchtlim 26055 | . . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1 | |
40 | rlimo1 14973 | . . . . . 6 ⊢ ((𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1 → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ∈ 𝑂(1)) | |
41 | 39, 40 | ax-mp 5 | . . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ∈ 𝑂(1) |
42 | o1mul 14971 | . . . . 5 ⊢ (((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) ∈ 𝑂(1)) | |
43 | 38, 41, 42 | sylancl 588 | . . . 4 ⊢ (⊤ → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) ∈ 𝑂(1)) |
44 | 34, 43 | eqeltrd 2913 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1)) |
45 | rerpdivcl 12420 | . . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ ℝ) | |
46 | 18, 45 | mpancom 686 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) / 𝑥) ∈ ℝ) |
47 | 46 | recnd 10669 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) / 𝑥) ∈ ℂ) |
48 | 47 | adantl 484 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ ℂ) |
49 | 48 | fmpttd 6879 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)):ℝ+⟶ℂ) |
50 | rpssre 12397 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
51 | 50 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
52 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → 2 ∈ ℝ) |
53 | 49, 51, 52 | o1resb 14923 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1))) |
54 | 44, 53 | mpbird 259 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
55 | 54 | mptru 1544 | 1 ⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 class class class wbr 5066 ↦ cmpt 5146 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 +∞cpnf 10672 < clt 10675 ≤ cle 10676 / cdiv 11297 2c2 11693 ℝ+crp 12390 [,)cico 12741 ⇝𝑟 crli 14842 𝑂(1)co1 14843 θccht 25668 ψcchp 25670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-o1 14847 df-lo1 14848 df-sum 15043 df-ef 15421 df-e 15422 df-sin 15423 df-cos 15424 df-pi 15426 df-dvds 15608 df-gcd 15844 df-prm 16016 df-pc 16174 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141 df-cht 25674 df-vma 25675 df-chp 25676 df-ppi 25677 |
This theorem is referenced by: chpo1ubb 26057 vmadivsum 26058 selberg2lem 26126 pntrmax 26140 pntrsumo1 26141 pntrlog2bndlem2 26154 |
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