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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 11790 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
2 | elicopnf 12919 | . . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))) | |
3 | 1, 2 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)) |
4 | chtrpcl 25912 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ+) | |
5 | 3, 4 | sylbi 220 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ+) |
6 | 5 | rpcnne0d 12523 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0)) |
7 | 3 | simplbi 501 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ) |
8 | 0red 10722 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ) | |
9 | 1 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ) |
10 | 2pos 11819 | . . . . . . . . . . . 12 ⊢ 0 < 2 | |
11 | 10 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2) |
12 | 3 | simprbi 500 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥) |
13 | 8, 9, 7, 11, 12 | ltletrd 10878 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥) |
14 | 7, 13 | elrpd 12511 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ+) |
15 | 14 | rpcnne0d 12523 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) |
16 | rpre 12480 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
17 | chpcl 25861 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ) | |
18 | 16, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ) |
19 | 18 | recnd 10747 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ) |
20 | 14, 19 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ) |
21 | dmdcan 11428 | . . . . . . . 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 1372 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥)) |
23 | 22 | adantl 485 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥)) |
24 | 23 | mpteq2dva 5125 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥))) |
25 | ovexd 7205 | . . . . . 6 ⊢ (⊤ → (2[,)+∞) ∈ V) | |
26 | ovexd 7205 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((θ‘𝑥) / 𝑥) ∈ V) | |
27 | ovexd 7205 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ V) | |
28 | eqidd 2739 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))) | |
29 | eqidd 2739 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) | |
30 | 25, 26, 27, 28, 29 | offval2 7444 | . . . . 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 3881 | . . . . . 6 ⊢ (2[,)+∞) ⊆ ℝ+ |
32 | resmpt 5879 | . . . . . 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 2784 | . . . 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 26212 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1) | |
37 | 36 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
38 | 35, 37 | o1res2 15010 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
39 | chpchtlim 26215 | . . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝𝑟 1 | |
40 | rlimo1 15064 | . . . . . 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 15062 | . . . . 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 589 | . . . 4 ⊢ (⊤ → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) ∈ 𝑂(1)) |
44 | 34, 43 | eqeltrd 2833 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1)) |
45 | rerpdivcl 12502 | . . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ ℝ) | |
46 | 18, 45 | mpancom 688 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) / 𝑥) ∈ ℝ) |
47 | 46 | recnd 10747 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → ((ψ‘𝑥) / 𝑥) ∈ ℂ) |
48 | 47 | adantl 485 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ ℂ) |
49 | 48 | fmpttd 6889 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)):ℝ+⟶ℂ) |
50 | rpssre 12479 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
51 | 50 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
52 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → 2 ∈ ℝ) |
53 | 49, 51, 52 | o1resb 15013 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1))) |
54 | 44, 53 | mpbird 260 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)) |
55 | 54 | mptru 1549 | 1 ⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ⊆ wss 3843 class class class wbr 5030 ↦ cmpt 5110 ↾ cres 5527 ‘cfv 6339 (class class class)co 7170 ∘f cof 7423 ℂcc 10613 ℝcr 10614 0cc0 10615 1c1 10616 · cmul 10620 +∞cpnf 10750 < clt 10753 ≤ cle 10754 / cdiv 11375 2c2 11771 ℝ+crp 12472 [,)cico 12823 ⇝𝑟 crli 14932 𝑂(1)co1 14933 θccht 25828 ψcchp 25830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-oadd 8135 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-ioo 12825 df-ioc 12826 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-fac 13726 df-bc 13755 df-hash 13783 df-shft 14516 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-limsup 14918 df-clim 14935 df-rlim 14936 df-o1 14937 df-lo1 14938 df-sum 15136 df-ef 15513 df-e 15514 df-sin 15515 df-cos 15516 df-pi 15518 df-dvds 15700 df-gcd 15938 df-prm 16113 df-pc 16274 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-perf 21888 df-cn 21978 df-cnp 21979 df-haus 22066 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cncf 23630 df-limc 24618 df-dv 24619 df-log 25300 df-cxp 25301 df-cht 25834 df-vma 25835 df-chp 25836 df-ppi 25837 |
This theorem is referenced by: chpo1ubb 26217 vmadivsum 26218 selberg2lem 26286 pntrmax 26300 pntrsumo1 26301 pntrlog2bndlem2 26314 |
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