chpo1ub - Metamath Proof Explorer

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Theorem chpo1ub 27418

Description: The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Assertion**
Ref	Expression
chpo1ub	⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)

**Proof of Theorem chpo1ub**
Step	Hyp	Ref	Expression
1		2re 12199	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
2		elicopnf 13345	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chtrpcl 27112	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
5	3, 4	sylbi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
6	5	rpcnne0d 12943	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
7	3	simplbi 497	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ)
8		0red 11115	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
9	1	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
10		2pos 12228	. . . . . . . . . . . 12 ⊢ 0 < 2
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2)
12	3	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥)
13	8, 9, 7, 11, 12	ltletrd 11273	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
14	7, 13	elrpd 12931	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
15	14	rpcnne0d 12943	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
16		rpre 12899	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
17		chpcl 27061	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
19	18	recnd 11140	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
20	14, 19	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
21		dmdcan 11831	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥))
23	22	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥))
24	23	mpteq2dva 5182	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)))
25		ovexd 7381	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ∈ V)
26		ovexd 7381	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((θ‘𝑥) / 𝑥) ∈ V)
27		ovexd 7381	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ V)
28		eqidd 2732	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)))
29		eqidd 2732	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))))
30	25, 26, 27, 28, 29	offval2 7630	. . . . 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 3933	. . . . . 6 ⊢ (2[,)+∞) ⊆ ℝ⁺
32		resmpt 5985	. . . . . 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 2777	. . . 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 27414	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
37	36	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
38	35, 37	o1res2 15470	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
39		chpchtlim 27417	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝_𝑟 1
40		rlimo1 15524	. . . . . 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 15522	. . . . 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 586	. . . 4 ⊢ (⊤ → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) ∈ 𝑂(1))
44	34, 43	eqeltrd 2831	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1))
45		rerpdivcl 12922	. . . . . . . 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 11140	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
48	47	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
49	48	fmpttd 7048	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
50		rpssre 12898	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
51	50	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
52	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
53	49, 51, 52	o1resb 15473	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ↔ ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1)))
54	44, 53	mpbird 257	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
55	54	mptru 1548	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ≠ wne 2928 Vcvv 3436 ⊆ wss 3897 class class class wbr 5089 ↦ cmpt 5170 ↾ cres 5616 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 · cmul 11011 +∞cpnf 11143 < clt 11146 ≤ cle 11147 / cdiv 11774 2c2 12180 ℝ⁺crp 12890 [,)cico 13247 ⇝_𝑟 crli 15392 𝑂(1)co1 15393 θccht 27028 ψcchp 27030

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-xnn0 12455 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-o1 15397 df-lo1 15398 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-pi 15979 df-dvds 16164 df-gcd 16406 df-prm 16583 df-pc 16749 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492 df-cxp 26493 df-cht 27034 df-vma 27035 df-chp 27036 df-ppi 27037

This theorem is referenced by: chpo1ubb 27419 vmadivsum 27420 selberg2lem 27488 pntrmax 27502 pntrsumo1 27503 pntrlog2bndlem2 27516

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