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Mirrors > Home > MPE Home > Th. List > cos1bnd | Structured version Visualization version 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 14098 | . . . . . . . 8 ⊢ (1↑2) = 1 | |
2 | 1 | oveq1i 7366 | . . . . . . 7 ⊢ ((1↑2) / 3) = (1 / 3) |
3 | 2 | oveq2i 7367 | . . . . . 6 ⊢ (2 · ((1↑2) / 3)) = (2 · (1 / 3)) |
4 | 2cn 12227 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 3cn 12233 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
6 | 3ne0 12258 | . . . . . . 7 ⊢ 3 ≠ 0 | |
7 | 4, 5, 6 | divreci 11899 | . . . . . 6 ⊢ (2 / 3) = (2 · (1 / 3)) |
8 | 3, 7 | eqtr4i 2767 | . . . . 5 ⊢ (2 · ((1↑2) / 3)) = (2 / 3) |
9 | 8 | oveq2i 7367 | . . . 4 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 − (2 / 3)) |
10 | ax-1cn 11108 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 4, 5, 6 | divcli 11896 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
12 | 5, 6 | reccli 11884 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
13 | df-3 12216 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
14 | 13 | oveq1i 7366 | . . . . . 6 ⊢ (3 / 3) = ((2 + 1) / 3) |
15 | 5, 6 | dividi 11887 | . . . . . 6 ⊢ (3 / 3) = 1 |
16 | 4, 10, 5, 6 | divdiri 11911 | . . . . . 6 ⊢ ((2 + 1) / 3) = ((2 / 3) + (1 / 3)) |
17 | 14, 15, 16 | 3eqtr3ri 2773 | . . . . 5 ⊢ ((2 / 3) + (1 / 3)) = 1 |
18 | 10, 11, 12, 17 | subaddrii 11489 | . . . 4 ⊢ (1 − (2 / 3)) = (1 / 3) |
19 | 9, 18 | eqtri 2764 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 / 3) |
20 | 1re 11154 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | 0lt1 11676 | . . . . 5 ⊢ 0 < 1 | |
22 | 1le1 11782 | . . . . 5 ⊢ 1 ≤ 1 | |
23 | 0xr 11201 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
24 | elioc2 13326 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1))) | |
25 | 23, 20, 24 | mp2an 690 | . . . . . 6 ⊢ (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1)) |
26 | cos01bnd 16067 | . . . . . 6 ⊢ (1 ∈ (0(,]1) → ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3)))) | |
27 | 25, 26 | sylbir 234 | . . . . 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 1461 | . . . 4 ⊢ ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3))) |
29 | 28 | simpli 484 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) < (cos‘1) |
30 | 19, 29 | eqbrtrri 5128 | . 2 ⊢ (1 / 3) < (cos‘1) |
31 | 28 | simpri 486 | . . 3 ⊢ (cos‘1) < (1 − ((1↑2) / 3)) |
32 | 2 | oveq2i 7367 | . . . 4 ⊢ (1 − ((1↑2) / 3)) = (1 − (1 / 3)) |
33 | 10, 12, 11 | subadd2i 11488 | . . . . 5 ⊢ ((1 − (1 / 3)) = (2 / 3) ↔ ((2 / 3) + (1 / 3)) = 1) |
34 | 17, 33 | mpbir 230 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
35 | 32, 34 | eqtri 2764 | . . 3 ⊢ (1 − ((1↑2) / 3)) = (2 / 3) |
36 | 31, 35 | breqtri 5130 | . 2 ⊢ (cos‘1) < (2 / 3) |
37 | 30, 36 | pm3.2i 471 | 1 ⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7356 ℝcr 11049 0cc0 11050 1c1 11051 + caddc 11053 · cmul 11055 ℝ*cxr 11187 < clt 11188 ≤ cle 11189 − cmin 11384 / cdiv 11811 2c2 12207 3c3 12208 (,]cioc 13264 ↑cexp 13966 cosccos 15946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-n0 12413 df-z 12499 df-uz 12763 df-rp 12915 df-ioc 13268 df-ico 13269 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-fac 14173 df-hash 14230 df-shft 14951 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-limsup 15352 df-clim 15369 df-rlim 15370 df-sum 15570 df-ef 15949 df-cos 15952 |
This theorem is referenced by: cos2bnd 16069 |
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