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Theorem pnt2 27642

Description: The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.)

**Assertion**
Ref	Expression
pnt2	⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1

**Proof of Theorem pnt2**
Step	Hyp	Ref	Expression
1		2re 12338	. . . . . . . . 9 ⊢ 2 ∈ ℝ
2		elicopnf 13476	. . . . . . . . 9 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chprpcl 27236	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ⁺)
5	3, 4	sylbi 216	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℝ⁺)
6	3	simplbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ)
7		0red 11267	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
8	1	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
9		2pos 12367	. . . . . . . . . 10 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2)
11	3	simprbi 495	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥)
12	7, 8, 6, 10, 11	ltletrd 11424	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
13	6, 12	elrpd 13067	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
14	5, 13	rpdivcld 13087	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
15	14	adantl 480	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
16		chtrpcl 27203	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
17	3, 16	sylbi 216	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
18	5, 17	rpdivcld 13087	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
19	18	adantl 480	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
20	13	ssriv 3983	. . . . . . 7 ⊢ (2[,)+∞) ⊆ ℝ⁺
21	20	a1i 11	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ⊆ ℝ⁺)
22		pnt3 27641	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1
23	22	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
24	21, 23	rlimres2 15563	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
25		chpchtlim 27508	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝_𝑟 1
26	25	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / (θ‘𝑥))) ⇝_𝑟 1)
27		ax-1ne0 11227	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	19	rpne0d 13075	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≠ 0)
30	15, 19, 24, 26, 28, 29	rlimdiv 15650	. . . 4 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) ⇝_𝑟 (1 / 1))
31		rpre 13036	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
32		chpcl 27152	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
34	33	recnd 11292	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
35	13, 34	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
36	13	rpcnne0d 13079	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
37	5	rpcnne0d 13079	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0))
38	17	rpcnne0d 13079	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
39		divdivdiv 11966	. . . . . . . 8 ⊢ ((((ψ‘𝑥) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) ∧ (((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0) ∧ ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))))
40	35, 36, 37, 38, 39	syl22anc 837	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))))
41	6	recnd 11292	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℂ)
42	41, 35	mulcomd 11285	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 · (ψ‘𝑥)) = ((ψ‘𝑥) · 𝑥))
43	42	oveq2d 7440	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)))
44		chtcl 27137	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (θ‘𝑥) ∈ ℝ)
45	31, 44	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℝ)
46	45	recnd 11292	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℂ)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℂ)
48		divcan5 11967	. . . . . . . 8 ⊢ (((θ‘𝑥) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ ((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0)) → (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)) = ((θ‘𝑥) / 𝑥))
49	47, 36, 37, 48	syl3anc 1368	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)) = ((θ‘𝑥) / 𝑥))
50	40, 43, 49	3eqtrd 2770	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = ((θ‘𝑥) / 𝑥))
51	50	mpteq2ia 5256	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))
52		resmpt 6046	. . . . . 6 ⊢ ((2[,)+∞) ⊆ ℝ⁺ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)))
53	20, 52	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))
54	51, 53	eqtr4i 2757	. . . 4 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞))
55		1div1e1 11955	. . . 4 ⊢ (1 / 1) = 1
56	30, 54, 55	3brtr3g 5186	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝_𝑟 1)
57		rerpdivcl 13058	. . . . . . . 8 ⊢ (((θ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℝ)
58	45, 57	mpancom 686	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((θ‘𝑥) / 𝑥) ∈ ℝ)
59	58	adantl 480	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℝ)
60	59	recnd 11292	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℂ)
61	60	fmpttd 7129	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
62		rpssre 13035	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
63	62	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
64	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
65	61, 63, 64	rlimresb 15567	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1 ↔ ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝_𝑟 1))
66	56, 65	mpbird 256	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1)
67	66	mptru 1541	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 class class class wbr 5153 ↦ cmpt 5236 ↾ cres 5684 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 · cmul 11163 +∞cpnf 11295 < clt 11298 ≤ cle 11299 / cdiv 11921 2c2 12319 ℝ⁺crp 13028 [,)cico 13380 ⇝_𝑟 crli 15487 θccht 27119 ψcchp 27121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-disj 5119 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12597 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-o1 15492 df-lo1 15493 df-sum 15691 df-ef 16069 df-e 16070 df-sin 16071 df-cos 16072 df-tan 16073 df-pi 16074 df-dvds 16257 df-gcd 16495 df-prm 16673 df-pc 16839 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-cmp 23382 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 df-ulm 26406 df-log 26583 df-cxp 26584 df-atan 26895 df-em 27021 df-cht 27125 df-vma 27126 df-chp 27127 df-ppi 27128 df-mu 27129

This theorem is referenced by: pnt 27643

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