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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 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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