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Theorem pnt2 27576

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 12255	. . . . . . . . 9 ⊢ 2 ∈ ℝ
2		elicopnf 13398	. . . . . . . . 9 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
3	1, 2	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
4		chprpcl 27170	. . . . . . . 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 11147	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ)
8	1	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ)
9		2pos 12284	. . . . . . . . . 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 11306	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥)
13	6, 12	elrpd 12983	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺)
14	5, 13	rpdivcld 13003	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
15	14	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ⁺)
16		chtrpcl 27138	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ⁺)
17	3, 16	sylbi 217	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ⁺)
18	5, 17	rpdivcld 13003	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
19	18	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ⁺)
20	13	ssriv 3925	. . . . . . 7 ⊢ (2[,)+∞) ⊆ ℝ⁺
21	20	a1i 11	. . . . . 6 ⊢ (⊤ → (2[,)+∞) ⊆ ℝ⁺)
22		pnt3 27575	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1
23	22	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
24	21, 23	rlimres2 15523	. . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ⇝_𝑟 1)
25		chpchtlim 27442	. . . . . 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 11107	. . . . . 6 ⊢ 1 ≠ 0
28	27	a1i 11	. . . . 5 ⊢ (⊤ → 1 ≠ 0)
29	19	rpne0d 12991	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≠ 0)
30	15, 19, 24, 26, 28, 29	rlimdiv 15608	. . . 4 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) ⇝_𝑟 (1 / 1))
31		rpre 12951	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
32		chpcl 27087	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
33	31, 32	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
34	33	recnd 11173	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
35	13, 34	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ)
36	13	rpcnne0d 12995	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
37	5	rpcnne0d 12995	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0))
38	17	rpcnne0d 12995	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0))
39		divdivdiv 11856	. . . . . . . 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 11173	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℂ)
42	41, 35	mulcomd 11166	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 · (ψ‘𝑥)) = ((ψ‘𝑥) · 𝑥))
43	42	oveq2d 7383	. . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)))
44		chtcl 27072	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (θ‘𝑥) ∈ ℝ)
45	31, 44	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℝ)
46	45	recnd 11173	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (θ‘𝑥) ∈ ℂ)
47	13, 46	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℂ)
48		divcan5 11857	. . . . . . . 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 2775	. . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = ((θ‘𝑥) / 𝑥))
51	50	mpteq2ia 5180	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥))
52		resmpt 6002	. . . . . 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 2762	. . . 4 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞))
55		1div1e1 11845	. . . 4 ⊢ (1 / 1) = 1
56	30, 54, 55	3brtr3g 5118	. . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝_𝑟 1)
57		rerpdivcl 12974	. . . . . . . 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 11173	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((θ‘𝑥) / 𝑥) ∈ ℂ)
61	60	fmpttd 7067	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((θ‘𝑥) / 𝑥)):ℝ⁺⟶ℂ)
62		rpssre 12950	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
63	62	a1i 11	. . . 4 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
64	1	a1i 11	. . . 4 ⊢ (⊤ → 2 ∈ ℝ)
65	61, 63, 64	rlimresb 15527	. . 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 2932 ⊆ wss 3889 class class class wbr 5085 ↦ cmpt 5166 ↾ cres 5633 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 +∞cpnf 11176 < clt 11179 ≤ cle 11180 / cdiv 11807 2c2 12236 ℝ⁺crp 12942 [,)cico 13300 ⇝_𝑟 crli 15447 θccht 27054 ψcchp 27056

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-tan 16036 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-ulm 26342 df-log 26520 df-cxp 26521 df-atan 26831 df-em 26956 df-cht 27060 df-vma 27061 df-chp 27062 df-ppi 27063 df-mu 27064

This theorem is referenced by: pnt 27577

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