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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 13408 | . . . . . . . 8 ⊢ (1↑2) = 1 | |
2 | 1 | oveq1i 7026 | . . . . . . 7 ⊢ ((1↑2) / 3) = (1 / 3) |
3 | 2 | oveq2i 7027 | . . . . . 6 ⊢ (2 · ((1↑2) / 3)) = (2 · (1 / 3)) |
4 | 2cn 11560 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 3cn 11566 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
6 | 3ne0 11591 | . . . . . . 7 ⊢ 3 ≠ 0 | |
7 | 4, 5, 6 | divreci 11233 | . . . . . 6 ⊢ (2 / 3) = (2 · (1 / 3)) |
8 | 3, 7 | eqtr4i 2822 | . . . . 5 ⊢ (2 · ((1↑2) / 3)) = (2 / 3) |
9 | 8 | oveq2i 7027 | . . . 4 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 − (2 / 3)) |
10 | ax-1cn 10441 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 4, 5, 6 | divcli 11230 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
12 | 5, 6 | reccli 11218 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
13 | df-3 11549 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
14 | 13 | oveq1i 7026 | . . . . . 6 ⊢ (3 / 3) = ((2 + 1) / 3) |
15 | 5, 6 | dividi 11221 | . . . . . 6 ⊢ (3 / 3) = 1 |
16 | 4, 10, 5, 6 | divdiri 11245 | . . . . . 6 ⊢ ((2 + 1) / 3) = ((2 / 3) + (1 / 3)) |
17 | 14, 15, 16 | 3eqtr3ri 2828 | . . . . 5 ⊢ ((2 / 3) + (1 / 3)) = 1 |
18 | 10, 11, 12, 17 | subaddrii 10823 | . . . 4 ⊢ (1 − (2 / 3)) = (1 / 3) |
19 | 9, 18 | eqtri 2819 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 / 3) |
20 | 1re 10487 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | 0lt1 11010 | . . . . 5 ⊢ 0 < 1 | |
22 | 1le1 11116 | . . . . 5 ⊢ 1 ≤ 1 | |
23 | 0xr 10534 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
24 | elioc2 12649 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1))) | |
25 | 23, 20, 24 | mp2an 688 | . . . . . 6 ⊢ (1 ∈ (0(,]1) ↔ (1 ∈ ℝ ∧ 0 < 1 ∧ 1 ≤ 1)) |
26 | cos01bnd 15372 | . . . . . 6 ⊢ (1 ∈ (0(,]1) → ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3)))) | |
27 | 25, 26 | sylbir 236 | . . . . 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 1453 | . . . 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 4985 | . 2 ⊢ (1 / 3) < (cos‘1) |
31 | 28 | simpri 486 | . . 3 ⊢ (cos‘1) < (1 − ((1↑2) / 3)) |
32 | 2 | oveq2i 7027 | . . . 4 ⊢ (1 − ((1↑2) / 3)) = (1 − (1 / 3)) |
33 | 10, 12, 11 | subadd2i 10822 | . . . . 5 ⊢ ((1 − (1 / 3)) = (2 / 3) ↔ ((2 / 3) + (1 / 3)) = 1) |
34 | 17, 33 | mpbir 232 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
35 | 32, 34 | eqtri 2819 | . . 3 ⊢ (1 − ((1↑2) / 3)) = (2 / 3) |
36 | 31, 35 | breqtri 4987 | . 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 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 − cmin 10717 / cdiv 11145 2c2 11540 3c3 11541 (,]cioc 12589 ↑cexp 13279 cosccos 15251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-ioc 12593 df-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-cos 15257 |
This theorem is referenced by: cos2bnd 15374 |
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