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Theorem pnt2 27588

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 12244	. . . . . . . . 9 ⊢ 2 ∈ ℝ
2		elicopnf 13387	. . . . . . . . 9 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chprpcl 27182	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ⁺)
5	3, 4	sylbi 217	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℝ⁺)
6	3	simplbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ)
7		0red 11136	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
8	1	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
9		2pos 12273	. . . . . . . . . 10 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2)
11	3	simprbi 497	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥)
12	7, 8, 6, 10, 11	ltletrd 11295	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
13	6, 12	elrpd 12972	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
14	5, 13	rpdivcld 12992	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
15	14	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
16		chtrpcl 27150	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
17	3, 16	sylbi 217	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
18	5, 17	rpdivcld 12992	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
19	18	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
20	13	ssriv 3926	. . . . . . 7 ⊢ (2[,)+∞) ⊆ ℝ⁺
21	20	a1i 11	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ⊆ ℝ⁺)
22		pnt3 27587	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1
23	22	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
24	21, 23	rlimres2 15512	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
25		chpchtlim 27454	. . . . . 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 11096	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	19	rpne0d 12980	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≠ 0)
30	15, 19, 24, 26, 28, 29	rlimdiv 15597	. . . 4 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) ⇝_𝑟 (1 / 1))
31		rpre 12940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
32		chpcl 27099	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
34	33	recnd 11162	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
35	13, 34	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
36	13	rpcnne0d 12984	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
37	5	rpcnne0d 12984	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0))
38	17	rpcnne0d 12984	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
39		divdivdiv 11845	. . . . . . . 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 839	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))))
41	6	recnd 11162	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℂ)
42	41, 35	mulcomd 11155	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 · (ψ‘𝑥)) = ((ψ‘𝑥) · 𝑥))
43	42	oveq2d 7374	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)))
44		chtcl 27084	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (θ‘𝑥) ∈ ℝ)
45	31, 44	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℝ)
46	45	recnd 11162	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℂ)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℂ)
48		divcan5 11846	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)) = ((θ‘𝑥) / 𝑥))
50	40, 43, 49	3eqtrd 2776	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = ((θ‘𝑥) / 𝑥))
51	50	mpteq2ia 5181	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))
52		resmpt 5994	. . . . . 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 2763	. . . 4 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞))
55		1div1e1 11834	. . . 4 ⊢ (1 / 1) = 1
56	30, 54, 55	3brtr3g 5119	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝_𝑟 1)
57		rerpdivcl 12963	. . . . . . . 8 ⊢ (((θ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℝ)
58	45, 57	mpancom 689	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((θ‘𝑥) / 𝑥) ∈ ℝ)
59	58	adantl 481	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℝ)
60	59	recnd 11162	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℂ)
61	60	fmpttd 7059	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
62		rpssre 12939	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
63	62	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
64	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
65	61, 63, 64	rlimresb 15516	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1 ↔ ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝_𝑟 1))
66	56, 65	mpbird 257	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1)
67	66	mptru 1549	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ⇝_𝑟 1

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ↾ cres 5624 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 · cmul 11032 +∞cpnf 11165 < clt 11168 ≤ cle 11169 / cdiv 11796 2c2 12225 ℝ⁺crp 12931 [,)cico 13289 ⇝_𝑟 crli 15436 θccht 27066 ψcchp 27068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-xnn0 12500 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-o1 15441 df-lo1 15442 df-sum 15638 df-ef 16021 df-e 16022 df-sin 16023 df-cos 16024 df-tan 16025 df-pi 16026 df-dvds 16211 df-gcd 16453 df-prm 16630 df-pc 16797 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21334 df-xmet 21335 df-met 21336 df-bl 21337 df-mopn 21338 df-fbas 21339 df-fg 21340 df-cnfld 21343 df-top 22867 df-topon 22884 df-topsp 22906 df-bases 22919 df-cld 22992 df-ntr 22993 df-cls 22994 df-nei 23071 df-lp 23109 df-perf 23110 df-cn 23200 df-cnp 23201 df-haus 23288 df-cmp 23360 df-tx 23535 df-hmeo 23728 df-fil 23819 df-fm 23911 df-flim 23912 df-flf 23913 df-xms 24293 df-ms 24294 df-tms 24295 df-cncf 24853 df-limc 25841 df-dv 25842 df-ulm 26353 df-log 26531 df-cxp 26532 df-atan 26842 df-em 26968 df-cht 27072 df-vma 27073 df-chp 27074 df-ppi 27075 df-mu 27076

This theorem is referenced by: pnt 27589

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