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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 11518 | . 2 ⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) | |
2 | lttri3 7868 | . . . . 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 8814 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | pire 12915 | . . . . . 6 ⊢ π ∈ ℝ | |
6 | 4, 5 | remulcli 7804 | . . . . 5 ⊢ (2 · π) ∈ ℝ |
7 | 6 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ ℝ) |
8 | 2rp 9475 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
9 | pirp 12918 | . . . . . . 7 ⊢ π ∈ ℝ+ | |
10 | rpmulcl 9495 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ π ∈ ℝ+) → (2 · π) ∈ ℝ+) | |
11 | 8, 9, 10 | mp2an 423 | . . . . . 6 ⊢ (2 · π) ∈ ℝ+ |
12 | 6 | recni 7802 | . . . . . . 7 ⊢ (2 · π) ∈ ℂ |
13 | cos2pi 12933 | . . . . . . 7 ⊢ (cos‘(2 · π)) = 1 | |
14 | cosf 11448 | . . . . . . . . 9 ⊢ cos:ℂ⟶ℂ | |
15 | ffn 5280 | . . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . 8 ⊢ cos Fn ℂ |
17 | fniniseg 5548 | . . . . . . . 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 927 | . . . . . 6 ⊢ (2 · π) ∈ (◡cos “ {1}) |
20 | 11, 19 | elini 3265 | . . . . 5 ⊢ (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1})) |
21 | 20 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1}))) |
22 | elinel2 3268 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ (◡cos “ {1})) | |
23 | fniniseg 5548 | . . . . . . . . . . 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 3267 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ+) | |
29 | 28 | rpred 9513 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ) |
30 | 29 | adantr 274 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ ℝ) |
31 | 28 | rpgt0d 9516 | . . . . . . . . . 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 7836 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ* | |
35 | 6 | rexri 7847 | . . . . . . . . . 10 ⊢ (2 · π) ∈ ℝ* |
36 | elioo2 9734 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ*) → (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π)))) | |
37 | 34, 35, 36 | mp2an 423 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π))) |
38 | 30, 32, 33, 37 | syl3anbrc 1166 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ (0(,)(2 · π))) |
39 | cos02pilt1 12980 | . . . . . . . 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 3960 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 < 1) |
42 | 1red 7805 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 ∈ ℝ) | |
43 | 42 | ltnrd 7899 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → ¬ 1 < 1) |
44 | 41, 43 | pm2.65da 651 | . . . . 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 6922 | . . 3 ⊢ (⊤ → inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π)) |
47 | 46 | mptru 1341 | . 2 ⊢ inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π) |
48 | 1, 47 | eqtri 2161 | 1 ⊢ τ = (2 · π) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ⊤wtru 1333 ∈ wcel 1481 ∩ cin 3075 {csn 3532 class class class wbr 3937 ◡ccnv 4546 “ cima 4550 Fn wfn 5126 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 infcinf 6878 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 · cmul 7649 ℝ*cxr 7823 < clt 7824 2c2 8795 ℝ+crp 9470 (,)cioo 9701 cosccos 11388 πcpi 11390 τctau 11517 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-tau 11518 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834 |
This theorem is referenced by: (None) |
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