![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cos1bnd | GIF version |
Description: Bounds on the cosine of 1. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
cos1bnd | ⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq1 10102 | . . . . . . . 8 ⊢ (1↑2) = 1 | |
2 | 1 | oveq1i 5676 | . . . . . . 7 ⊢ ((1↑2) / 3) = (1 / 3) |
3 | 2 | oveq2i 5677 | . . . . . 6 ⊢ (2 · ((1↑2) / 3)) = (2 · (1 / 3)) |
4 | 2cn 8547 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 3cn 8551 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
6 | 3ap0 8572 | . . . . . . 7 ⊢ 3 # 0 | |
7 | 4, 5, 6 | divrecapi 8278 | . . . . . 6 ⊢ (2 / 3) = (2 · (1 / 3)) |
8 | 3, 7 | eqtr4i 2112 | . . . . 5 ⊢ (2 · ((1↑2) / 3)) = (2 / 3) |
9 | 8 | oveq2i 5677 | . . . 4 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 − (2 / 3)) |
10 | ax-1cn 7492 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 4, 5, 6 | divclapi 8275 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
12 | 5, 6 | recclapi 8263 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
13 | df-3 8536 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
14 | 13 | oveq1i 5676 | . . . . . 6 ⊢ (3 / 3) = ((2 + 1) / 3) |
15 | 5, 6 | dividapi 8266 | . . . . . 6 ⊢ (3 / 3) = 1 |
16 | 4, 10, 5, 6 | divdirapi 8290 | . . . . . 6 ⊢ ((2 + 1) / 3) = ((2 / 3) + (1 / 3)) |
17 | 14, 15, 16 | 3eqtr3ri 2118 | . . . . 5 ⊢ ((2 / 3) + (1 / 3)) = 1 |
18 | 10, 11, 12, 17 | subaddrii 7825 | . . . 4 ⊢ (1 − (2 / 3)) = (1 / 3) |
19 | 9, 18 | eqtri 2109 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 / 3) |
20 | 1re 7541 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | 0lt1 7664 | . . . . 5 ⊢ 0 < 1 | |
22 | 1le1 8103 | . . . . 5 ⊢ 1 ≤ 1 | |
23 | 0xr 7588 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
24 | elioc2 9408 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1))) | |
25 | 23, 20, 24 | mp2an 418 | . . . . . 6 ⊢ (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1)) |
26 | cos01bnd 11103 | . . . . . 6 ⊢ (1 ∈ (0(,]1) → ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3)))) | |
27 | 25, 26 | sylbir 134 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1) → ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3)))) |
28 | 20, 21, 22, 27 | mp3an 1274 | . . . 4 ⊢ ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3))) |
29 | 28 | simpli 110 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) < (cos‘1) |
30 | 19, 29 | eqbrtrri 3872 | . 2 ⊢ (1 / 3) < (cos‘1) |
31 | 28 | simpri 112 | . . 3 ⊢ (cos‘1) < (1 − ((1↑2) / 3)) |
32 | 2 | oveq2i 5677 | . . . 4 ⊢ (1 − ((1↑2) / 3)) = (1 − (1 / 3)) |
33 | 10, 12, 11 | subadd2i 7824 | . . . . 5 ⊢ ((1 − (1 / 3)) = (2 / 3) ↔ ((2 / 3) + (1 / 3)) = 1) |
34 | 17, 33 | mpbir 145 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
35 | 32, 34 | eqtri 2109 | . . 3 ⊢ (1 − ((1↑2) / 3)) = (2 / 3) |
36 | 31, 35 | breqtri 3874 | . 2 ⊢ (cos‘1) < (2 / 3) |
37 | 30, 36 | pm3.2i 267 | 1 ⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 ‘cfv 5028 (class class class)co 5666 ℝcr 7403 0cc0 7404 1c1 7405 + caddc 7407 · cmul 7409 ℝ*cxr 7575 < clt 7576 ≤ cle 7577 − cmin 7707 / cdiv 8193 2c2 8527 3c3 8528 (,]cioc 9361 ↑cexp 10008 cosccos 10989 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-isom 5037 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-frec 6170 df-1o 6195 df-oadd 6199 df-er 6306 df-en 6512 df-dom 6513 df-fin 6514 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-5 8538 df-6 8539 df-7 8540 df-8 8541 df-n0 8728 df-z 8805 df-uz 9074 df-q 9159 df-rp 9189 df-ioc 9365 df-ico 9366 df-fz 9479 df-fzo 9608 df-iseq 9907 df-seq3 9908 df-exp 10009 df-fac 10188 df-ihash 10238 df-shft 10303 df-cj 10330 df-re 10331 df-im 10332 df-rsqrt 10485 df-abs 10486 df-clim 10721 df-isum 10797 df-ef 10992 df-cos 10995 |
This theorem is referenced by: cos2bnd 11105 |
Copyright terms: Public domain | W3C validator |