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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 11782 | . 2 ⊢ τ = inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) | |
2 | lttri3 8036 | . . . . 5 ⊢ ((𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) | |
3 | 2 | adantl 277 | . . . 4 ⊢ ((⊤ ∧ (𝑓 ∈ ℝ ∧ 𝑔 ∈ ℝ)) → (𝑓 = 𝑔 ↔ (¬ 𝑓 < 𝑔 ∧ ¬ 𝑔 < 𝑓))) |
4 | 2re 8988 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | pire 14177 | . . . . . 6 ⊢ π ∈ ℝ | |
6 | 4, 5 | remulcli 7970 | . . . . 5 ⊢ (2 · π) ∈ ℝ |
7 | 6 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ ℝ) |
8 | 2rp 9657 | . . . . . . 7 ⊢ 2 ∈ ℝ+ | |
9 | pirp 14180 | . . . . . . 7 ⊢ π ∈ ℝ+ | |
10 | rpmulcl 9677 | . . . . . . 7 ⊢ ((2 ∈ ℝ+ ∧ π ∈ ℝ+) → (2 · π) ∈ ℝ+) | |
11 | 8, 9, 10 | mp2an 426 | . . . . . 6 ⊢ (2 · π) ∈ ℝ+ |
12 | 6 | recni 7968 | . . . . . . 7 ⊢ (2 · π) ∈ ℂ |
13 | cos2pi 14195 | . . . . . . 7 ⊢ (cos‘(2 · π)) = 1 | |
14 | cosf 11712 | . . . . . . . . 9 ⊢ cos:ℂ⟶ℂ | |
15 | ffn 5365 | . . . . . . . . 9 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
16 | 14, 15 | ax-mp 5 | . . . . . . . 8 ⊢ cos Fn ℂ |
17 | fniniseg 5636 | . . . . . . . 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 942 | . . . . . 6 ⊢ (2 · π) ∈ (◡cos “ {1}) |
20 | 11, 19 | elini 3319 | . . . . 5 ⊢ (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1})) |
21 | 20 | a1i 9 | . . . 4 ⊢ (⊤ → (2 · π) ∈ (ℝ+ ∩ (◡cos “ {1}))) |
22 | elinel2 3322 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ (◡cos “ {1})) | |
23 | fniniseg 5636 | . . . . . . . . . . 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 122 | . . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → (𝑥 ∈ ℂ ∧ (cos‘𝑥) = 1)) |
26 | 25 | simprd 114 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → (cos‘𝑥) = 1) |
27 | 26 | adantr 276 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → (cos‘𝑥) = 1) |
28 | elinel1 3321 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ+) | |
29 | 28 | rpred 9695 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 𝑥 ∈ ℝ) |
30 | 29 | adantr 276 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ ℝ) |
31 | 28 | rpgt0d 9698 | . . . . . . . . . 10 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → 0 < 𝑥) |
32 | 31 | adantr 276 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 0 < 𝑥) |
33 | simpr 110 | . . . . . . . . 9 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 < (2 · π)) | |
34 | 0xr 8003 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ* | |
35 | 6 | rexri 8014 | . . . . . . . . . 10 ⊢ (2 · π) ∈ ℝ* |
36 | elioo2 9920 | . . . . . . . . . 10 ⊢ ((0 ∈ ℝ* ∧ (2 · π) ∈ ℝ*) → (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π)))) | |
37 | 34, 35, 36 | mp2an 426 | . . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)(2 · π)) ↔ (𝑥 ∈ ℝ ∧ 0 < 𝑥 ∧ 𝑥 < (2 · π))) |
38 | 30, 32, 33, 37 | syl3anbrc 1181 | . . . . . . . 8 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 𝑥 ∈ (0(,)(2 · π))) |
39 | cos02pilt1 14242 | . . . . . . . 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 4027 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 < 1) |
42 | 1red 7971 | . . . . . . 7 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → 1 ∈ ℝ) | |
43 | 42 | ltnrd 8068 | . . . . . 6 ⊢ ((𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) ∧ 𝑥 < (2 · π)) → ¬ 1 < 1) |
44 | 41, 43 | pm2.65da 661 | . . . . 5 ⊢ (𝑥 ∈ (ℝ+ ∩ (◡cos “ {1})) → ¬ 𝑥 < (2 · π)) |
45 | 44 | adantl 277 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (ℝ+ ∩ (◡cos “ {1}))) → ¬ 𝑥 < (2 · π)) |
46 | 3, 7, 21, 45 | infminti 7025 | . . 3 ⊢ (⊤ → inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π)) |
47 | 46 | mptru 1362 | . 2 ⊢ inf((ℝ+ ∩ (◡cos “ {1})), ℝ, < ) = (2 · π) |
48 | 1, 47 | eqtri 2198 | 1 ⊢ τ = (2 · π) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 ∩ cin 3128 {csn 3592 class class class wbr 4003 ◡ccnv 4625 “ cima 4629 Fn wfn 5211 ⟶wf 5212 ‘cfv 5216 (class class class)co 5874 infcinf 6981 ℂcc 7808 ℝcr 7809 0cc0 7810 1c1 7811 · cmul 7815 ℝ*cxr 7990 < clt 7991 2c2 8969 ℝ+crp 9652 (,)cioo 9887 cosccos 11652 πcpi 11654 τctau 11781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 ax-pre-suploc 7931 ax-addf 7932 ax-mulf 7933 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-disj 3981 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-of 6082 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-map 6649 df-pm 6650 df-en 6740 df-dom 6741 df-fin 6742 df-sup 6982 df-inf 6983 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-5 8980 df-6 8981 df-7 8982 df-8 8983 df-9 8984 df-n0 9176 df-z 9253 df-uz 9528 df-q 9619 df-rp 9653 df-xneg 9771 df-xadd 9772 df-ioo 9891 df-ioc 9892 df-ico 9893 df-icc 9894 df-fz 10008 df-fzo 10142 df-seqfrec 10445 df-exp 10519 df-fac 10705 df-bc 10727 df-ihash 10755 df-shft 10823 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-clim 11286 df-sumdc 11361 df-ef 11655 df-sin 11657 df-cos 11658 df-pi 11660 df-tau 11782 df-rest 12689 df-topgen 12708 df-psmet 13417 df-xmet 13418 df-met 13419 df-bl 13420 df-mopn 13421 df-top 13468 df-topon 13481 df-bases 13513 df-ntr 13566 df-cn 13658 df-cnp 13659 df-tx 13723 df-cncf 14028 df-limced 14095 df-dvap 14096 |
This theorem is referenced by: (None) |
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