chpo1ub - Metamath Proof Explorer

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Theorem chpo1ub 27524

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 12340	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
2		elicopnf 13485	. . . . . . . . . . 11 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chtrpcl 27218	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
5	3, 4	sylbi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
6	5	rpcnne0d 13086	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
7	3	simplbi 497	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ)
8		0red 11264	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
9	1	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
10		2pos 12369	. . . . . . . . . . . 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 11421	. . . . . . . . . 10 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
14	7, 13	elrpd 13074	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
15	14	rpcnne0d 13086	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
16		rpre 13043	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
17		chpcl 27167	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
19	18	recnd 11289	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
20	14, 19	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
21		dmdcan 11977	. . . . . . . 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 5242	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((θ‘𝑥) / 𝑥) · ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)))
25		ovexd 7466	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ∈ V)
26		ovexd 7466	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((θ‘𝑥) / 𝑥) ∈ V)
27		ovexd 7466	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ V)
28		eqidd 2738	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)))
29		eqidd 2738	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))))
30	25, 26, 27, 28, 29	offval2 7717	. . . . 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 3987	. . . . . 6 ⊢ (2[,)+∞) ⊆ ℝ⁺
32		resmpt 6055	. . . . . 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 2788	. . . 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 27520	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1)
37	36	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
38	35, 37	o1res2 15599	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) ∈ 𝑂(1))
39		chpchtlim 27523	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝_𝑟 1
40		rlimo1 15653	. . . . . 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 15651	. . . . 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 2841	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ∈ 𝑂(1))
45		rerpdivcl 13065	. . . . . . . 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 11289	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
48	47	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
49	48	fmpttd 7135	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
50		rpssre 13042	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
51	50	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
52	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
53	49, 51, 52	o1resb 15602	. . 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 1547	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 ∘_f cof 7695 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 +∞cpnf 11292 < clt 11295 ≤ cle 11296 / cdiv 11920 2c2 12321 ℝ⁺crp 13034 [,)cico 13389 ⇝_𝑟 crli 15521 𝑂(1)co1 15522 θccht 27134 ψcchp 27136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-o1 15526 df-lo1 15527 df-sum 15723 df-ef 16103 df-e 16104 df-sin 16105 df-cos 16106 df-pi 16108 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-cht 27140 df-vma 27141 df-chp 27142 df-ppi 27143

This theorem is referenced by: chpo1ubb 27525 vmadivsum 27526 selberg2lem 27594 pntrmax 27608 pntrsumo1 27609 pntrlog2bndlem2 27622

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