Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > taupi | GIF version |
Description: Relationship between τ and π. This can be seen as connecting the ratio of a circle's circumference to its radius and the ratio of a circle's circumference to its diameter. (Contributed by Jim Kingdon, 19-Feb-2019.) (Revised by AV, 1-Oct-2020.) |
Ref | Expression |
---|---|
taupi | ⊢ τ = (2 · π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tau 11485 | . 2 ⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) | |
2 | lttri3 7847 | . . . . 5 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
3 | 2 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
4 | 2re 8793 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | pire 12870 | . . . . . 6 ⊢ π ∈ ℝ | |
6 | 4, 5 | remulcli 7783 | . . . . 5 ⊢ (2 · π) ∈ ℝ |
7 | 6 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ ℝ) |
8 | 2rp 9449 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
9 | pirp 12873 | . . . . . . 7 ⊢ π ∈ ℝ+ | |
10 | rpmulcl 9469 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ π ∈ ℝ+) → (2 · π) ∈ ℝ+) | |
11 | 8, 9, 10 | mp2an 422 | . . . . . 6 ⊢ (2 · π) ∈ ℝ+ |
12 | 6 | recni 7781 | . . . . . . 7 ⊢ (2 · π) ∈ ℂ |
13 | cos2pi 12888 | . . . . . . 7 ⊢ (cos‘(2 · π)) = 1 | |
14 | cosf 11415 | . . . . . . . . 9 ⊢ cos:ℂ⟶ℂ | |
15 | ffn 5272 | . . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . 8 ⊢ cos Fn ℂ |
17 | fniniseg 5540 | . . . . . . . 8 ⊢ (cos Fn ℂ → ((2 · π) ∈ (◡cos “ {1}) ↔ ((2 · π) ∈ ℂ ∧ (cos‘(2 · π)) = 1))) | |
18 | 16, 17 | ax-mp 5 | . . . . . . 7 ⊢ ((2 · π) ∈ (◡cos “ {1}) ↔ ((2 · π) ∈ ℂ ∧ (cos‘(2 · π)) = 1)) |
19 | 12, 13, 18 | mpbir2an 926 | . . . . . 6 ⊢ (2 · π) ∈ (◡cos “ {1}) |
20 | 11, 19 | elini 3260 | . . . . 5 ⊢ (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1})) |
21 | 20 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1}))) |
22 | elinel2 3263 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ (◡cos “ {1})) | |
23 | fniniseg 5540 | . . . . . . . . . . 11 ⊢ (cos Fn ℂ → (𝑥 ∈ (◡cos “ {1}) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) = 1))) | |
24 | 16, 23 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (◡cos “ {1}) ↔ (𝑥 ∈ ℂ ∧ (cos‘𝑥) = 1)) |
25 | 22, 24 | sylib 121 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → (𝑥 ∈ ℂ ∧ (cos‘𝑥) = 1)) |
26 | 25 | simprd 113 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → (cos‘𝑥) = 1) |
27 | 26 | adantr 274 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → (cos‘𝑥) = 1) |
28 | elinel1 3262 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ+) | |
29 | 28 | rpred 9486 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ) |
30 | 29 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ ℝ) |
31 | 28 | rpgt0d 9489 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 0 < 𝑥) |
32 | 31 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 0 < 𝑥) |
33 | simpr 109 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 < (2 · π)) | |
34 | 0xr 7815 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ* | |
35 | 6 | rexri 7826 | . . . . . . . . . 10 ⊢ (2 · π) ∈ ℝ* |
36 | elioo2 9707 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ*) → (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π)))) | |
37 | 34, 35, 36 | mp2an 422 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π))) |
38 | 30, 32, 33, 37 | syl3anbrc 1165 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ (0(,)(2 · π))) |
39 | cos02pilt1 12935 | . . . . . . . 8 ⊢ (𝑥 ∈ (0(,)(2 · π)) → (cos‘𝑥) < 1) | |
40 | 38, 39 | syl 14 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → (cos‘𝑥) < 1) |
41 | 27, 40 | eqbrtrrd 3952 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 < 1) |
42 | 1red 7784 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 ∈ ℝ) | |
43 | 42 | ltnrd 7878 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → ¬ 1 < 1) |
44 | 41, 43 | pm2.65da 650 | . . . . 5 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → ¬ 𝑥 < (2 · π)) |
45 | 44 | adantl 275 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ+ ∩ (◡cos “ {1}))) → ¬ 𝑥 < (2 · π)) |
46 | 3, 7, 21, 45 | infminti 6914 | . . 3 ⊢ (⊤ → inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π)) |
47 | 46 | mptru 1340 | . 2 ⊢ inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π) |
48 | 1, 47 | eqtri 2160 | 1 ⊢ τ = (2 · π) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 ∩ cin 3070 {csn 3527 class class class wbr 3929 ◡ccnv 4538 “ cima 4542 Fn wfn 5118 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 infcinf 6870 ℂcc 7621 ℝcr 7622 0cc0 7623 1c1 7624 · cmul 7628 ℝ*cxr 7802 < clt 7803 2c2 8774 ℝ+crp 9444 (,)cioo 9674 cosccos 11354 πcpi 11356 τctau 11484 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 ax-arch 7742 ax-caucvg 7743 ax-pre-suploc 7744 ax-addf 7745 ax-mulf 7746 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-map 6544 df-pm 6545 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-inf 6872 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 df-inn 8724 df-2 8782 df-3 8783 df-4 8784 df-5 8785 df-6 8786 df-7 8787 df-8 8788 df-9 8789 df-n0 8981 df-z 9058 df-uz 9330 df-q 9415 df-rp 9445 df-xneg 9562 df-xadd 9563 df-ioo 9678 df-ioc 9679 df-ico 9680 df-icc 9681 df-fz 9794 df-fzo 9923 df-seqfrec 10222 df-exp 10296 df-fac 10475 df-bc 10497 df-ihash 10525 df-shft 10590 df-cj 10617 df-re 10618 df-im 10619 df-rsqrt 10773 df-abs 10774 df-clim 11051 df-sumdc 11126 df-ef 11357 df-sin 11359 df-cos 11360 df-pi 11362 df-tau 11485 df-rest 12125 df-topgen 12144 df-psmet 12159 df-xmet 12160 df-met 12161 df-bl 12162 df-mopn 12163 df-top 12168 df-topon 12181 df-bases 12213 df-ntr 12268 df-cn 12360 df-cnp 12361 df-tx 12425 df-cncf 12730 df-limced 12797 df-dvap 12798 |
This theorem is referenced by: (None) |
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