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Mirrors > Home > MPE Home > Th. List > pnt2 | Structured version Visualization version GIF version |
Description: The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to 𝑥. (Contributed by Mario Carneiro, 1-Jun-2016.) |
Ref | Expression |
---|---|
pnt2 | ⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11517 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
2 | elicopnf 12652 | . . . . . . . . 9 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))) | |
3 | 1, 2 | ax-mp 5 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)) |
4 | chprpcl 25488 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (ψ‘𝑥) ∈ ℝ+) | |
5 | 3, 4 | sylbi 209 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℝ+) |
6 | 3 | simplbi 490 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ) |
7 | 0red 10445 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 ∈ ℝ) | |
8 | 1 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ∈ ℝ) |
9 | 2pos 11553 | . . . . . . . . . 10 ⊢ 0 < 2 | |
10 | 9 | a1i 11 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 2) |
11 | 3 | simprbi 489 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 2 ≤ 𝑥) |
12 | 7, 8, 6, 10, 11 | ltletrd 10602 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → 0 < 𝑥) |
13 | 6, 12 | elrpd 12248 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ+) |
14 | 5, 13 | rpdivcld 12268 | . . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / 𝑥) ∈ ℝ+) |
15 | 14 | adantl 474 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ+) |
16 | chtrpcl 25457 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (θ‘𝑥) ∈ ℝ+) | |
17 | 3, 16 | sylbi 209 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℝ+) |
18 | 5, 17 | rpdivcld 12268 | . . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ+) |
19 | 18 | adantl 474 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ∈ ℝ+) |
20 | 13 | ssriv 3864 | . . . . . . 7 ⊢ (2[,)+∞) ⊆ ℝ+ |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (⊤ → (2[,)+∞) ⊆ ℝ+) |
22 | pnt3 25893 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1 | |
23 | 22 | a1i 11 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1) |
24 | 21, 23 | rlimres2 14782 | . . . . 5 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ⇝𝑟 1) |
25 | chpchtlim 25760 | . . . . . 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 10406 | . . . . . 6 ⊢ 1 ≠ 0 | |
28 | 27 | a1i 11 | . . . . 5 ⊢ (⊤ → 1 ≠ 0) |
29 | 19 | rpne0d 12256 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (2[,)+∞)) → ((ψ‘𝑥) / (θ‘𝑥)) ≠ 0) |
30 | 15, 19, 24, 26, 28, 29 | rlimdiv 14866 | . . . 4 ⊢ (⊤ → (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) ⇝𝑟 (1 / 1)) |
31 | rpre 12215 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
32 | chpcl 25406 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ) | |
33 | 31, 32 | syl 17 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ) |
34 | 33 | recnd 10470 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ) |
35 | 13, 34 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (ψ‘𝑥) ∈ ℂ) |
36 | 13 | rpcnne0d 12260 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) |
37 | 5 | rpcnne0d 12260 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((ψ‘𝑥) ∈ ℂ ∧ (ψ‘𝑥) ≠ 0)) |
38 | 17 | rpcnne0d 12260 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → ((θ‘𝑥) ∈ ℂ ∧ (θ‘𝑥) ≠ 0)) |
39 | divdivdiv 11144 | . . . . . . . 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 826 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥)))) |
41 | 6 | recnd 10470 | . . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℂ) |
42 | 41, 35 | mulcomd 10463 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (𝑥 · (ψ‘𝑥)) = ((ψ‘𝑥) · 𝑥)) |
43 | 42 | oveq2d 6994 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / (𝑥 · (ψ‘𝑥))) = (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥))) |
44 | chtcl 25391 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (θ‘𝑥) ∈ ℝ) | |
45 | 31, 44 | syl 17 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → (θ‘𝑥) ∈ ℝ) |
46 | 45 | recnd 10470 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (θ‘𝑥) ∈ ℂ) |
47 | 13, 46 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ (2[,)+∞) → (θ‘𝑥) ∈ ℂ) |
48 | divcan5 11145 | . . . . . . . 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 1351 | . . . . . . 7 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) · (θ‘𝑥)) / ((ψ‘𝑥) · 𝑥)) = ((θ‘𝑥) / 𝑥)) |
50 | 40, 43, 49 | 3eqtrd 2818 | . . . . . 6 ⊢ (𝑥 ∈ (2[,)+∞) → (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥))) = ((θ‘𝑥) / 𝑥)) |
51 | 50 | mpteq2ia 5019 | . . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = (𝑥 ∈ (2[,)+∞) ↦ ((θ‘𝑥) / 𝑥)) |
52 | resmpt 5752 | . . . . . 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 2805 | . . . 4 ⊢ (𝑥 ∈ (2[,)+∞) ↦ (((ψ‘𝑥) / 𝑥) / ((ψ‘𝑥) / (θ‘𝑥)))) = ((𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) |
55 | 1div1e1 11133 | . . . 4 ⊢ (1 / 1) = 1 | |
56 | 30, 54, 55 | 3brtr3g 4963 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝𝑟 1) |
57 | rerpdivcl 12239 | . . . . . . . 8 ⊢ (((θ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → ((θ‘𝑥) / 𝑥) ∈ ℝ) | |
58 | 45, 57 | mpancom 675 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → ((θ‘𝑥) / 𝑥) ∈ ℝ) |
59 | 58 | adantl 474 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((θ‘𝑥) / 𝑥) ∈ ℝ) |
60 | 59 | recnd 10470 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ+) → ((θ‘𝑥) / 𝑥) ∈ ℂ) |
61 | 60 | fmpttd 6704 | . . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)):ℝ+⟶ℂ) |
62 | rpssre 12214 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
63 | 62 | a1i 11 | . . . 4 ⊢ (⊤ → ℝ+ ⊆ ℝ) |
64 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → 2 ∈ ℝ) |
65 | 61, 63, 64 | rlimresb 14786 | . . 3 ⊢ (⊤ → ((𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1 ↔ ((𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ↾ (2[,)+∞)) ⇝𝑟 1)) |
66 | 56, 65 | mpbird 249 | . 2 ⊢ (⊤ → (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1) |
67 | 66 | mptru 1514 | 1 ⊢ (𝑥 ∈ ℝ+ ↦ ((θ‘𝑥) / 𝑥)) ⇝𝑟 1 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 387 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 ≠ wne 2967 ⊆ wss 3831 class class class wbr 4930 ↦ cmpt 5009 ↾ cres 5410 ‘cfv 6190 (class class class)co 6978 ℂcc 10335 ℝcr 10336 0cc0 10337 1c1 10338 · cmul 10342 +∞cpnf 10473 < clt 10476 ≤ cle 10477 / cdiv 11100 2c2 11498 ℝ+crp 12207 [,)cico 12559 ⇝𝑟 crli 14706 θccht 25373 ψcchp 25375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-disj 4899 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-dju 9126 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-xnn0 11783 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-fac 13452 df-bc 13481 df-hash 13509 df-shft 14290 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-o1 14711 df-lo1 14712 df-sum 14907 df-ef 15284 df-e 15285 df-sin 15286 df-cos 15287 df-tan 15288 df-pi 15289 df-dvds 15471 df-gcd 15707 df-prm 15875 df-pc 16033 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-mulg 18015 df-cntz 18221 df-cmn 18671 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-xms 22636 df-ms 22637 df-tms 22638 df-cncf 23192 df-limc 24170 df-dv 24171 df-ulm 24671 df-log 24844 df-cxp 24845 df-atan 25149 df-em 25275 df-cht 25379 df-vma 25380 df-chp 25381 df-ppi 25382 df-mu 25383 |
This theorem is referenced by: pnt 25895 |
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