chpo1ub - Metamath Proof Explorer

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Theorem chpo1ub 26963

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 12282	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
2		elicopnf 13418	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chtrpcl 26659	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
5	3, 4	sylbi 216	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
6	5	rpcnne0d 13021	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
7	3	simplbi 499	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ)
8		0red 11213	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
9	1	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
10		2pos 12311	. . . . . . . . . . . 12 ⊢ 0 < 2
11	10	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2)
12	3	simprbi 498	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥)
13	8, 9, 7, 11, 12	ltletrd 11370	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
14	7, 13	elrpd 13009	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
15	14	rpcnne0d 13021	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
16		rpre 12978	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
17		chpcl 26608	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
19	18	recnd 11238	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
20	14, 19	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
21		dmdcan 11920	. . . . . . . 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 483	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥))) = ((ψ‘𝑥) / 𝑥))
24	23	mpteq2dva 5247	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)))
25		ovexd 7439	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ∈ V)
26		ovexd 7439	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((θ‘𝑥) / 𝑥) ∈ V)
27		ovexd 7439	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ V)
28		eqidd 2734	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)))
29		eqidd 2734	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))))
30	25, 26, 27, 28, 29	offval2 7685	. . . . 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 3985	. . . . . 6 ⊢ (2[,)+∞) ⊆ ℝ⁺
32		resmpt 6035	. . . . . 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 26959	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
37	36	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
38	35, 37	o1res2 15503	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
39		chpchtlim 26962	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝_𝑟 1
40		rlimo1 15557	. . . . . 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 15555	. . . . 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 587	. . . 4 ⊢ (⊤ → ((𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥)))) ∈ 𝑂(1))
44	34, 43	eqeltrd 2834	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1))
45		rerpdivcl 13000	. . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
46	18, 45	mpancom 687	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
47	46	recnd 11238	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
48	47	adantl 483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
49	48	fmpttd 7110	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
50		rpssre 12977	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
51	50	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
52	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
53	49, 51, 52	o1resb 15506	. . 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 1549	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ↾ cres 5677 ‘cfv 6540 (class class class)co 7404 ∘_f cof 7663 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 +∞cpnf 11241 < clt 11244 ≤ cle 11245 / cdiv 11867 2c2 12263 ℝ⁺crp 12970 [,)cico 13322 ⇝_𝑟 crli 15425 𝑂(1)co1 15426 θccht 26575 ψcchp 26577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-o1 15430 df-lo1 15431 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-lp 22622 df-perf 22623 df-cn 22713 df-cnp 22714 df-haus 22801 df-tx 23048 df-hmeo 23241 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-tms 23810 df-cncf 24376 df-limc 25365 df-dv 25366 df-log 26047 df-cxp 26048 df-cht 26581 df-vma 26582 df-chp 26583 df-ppi 26584

This theorem is referenced by: chpo1ubb 26964 vmadivsum 26965 selberg2lem 27033 pntrmax 27047 pntrsumo1 27048 pntrlog2bndlem2 27061

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