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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 13559 | . . . . . . . 8 ⊢ (1↑2) = 1 | |
2 | 1 | oveq1i 7166 | . . . . . . 7 ⊢ ((1↑2) / 3) = (1 / 3) |
3 | 2 | oveq2i 7167 | . . . . . 6 ⊢ (2 · ((1↑2) / 3)) = (2 · (1 / 3)) |
4 | 2cn 11713 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 3cn 11719 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
6 | 3ne0 11744 | . . . . . . 7 ⊢ 3 ≠ 0 | |
7 | 4, 5, 6 | divreci 11385 | . . . . . 6 ⊢ (2 / 3) = (2 · (1 / 3)) |
8 | 3, 7 | eqtr4i 2847 | . . . . 5 ⊢ (2 · ((1↑2) / 3)) = (2 / 3) |
9 | 8 | oveq2i 7167 | . . . 4 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 − (2 / 3)) |
10 | ax-1cn 10595 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 4, 5, 6 | divcli 11382 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
12 | 5, 6 | reccli 11370 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
13 | df-3 11702 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
14 | 13 | oveq1i 7166 | . . . . . 6 ⊢ (3 / 3) = ((2 + 1) / 3) |
15 | 5, 6 | dividi 11373 | . . . . . 6 ⊢ (3 / 3) = 1 |
16 | 4, 10, 5, 6 | divdiri 11397 | . . . . . 6 ⊢ ((2 + 1) / 3) = ((2 / 3) + (1 / 3)) |
17 | 14, 15, 16 | 3eqtr3ri 2853 | . . . . 5 ⊢ ((2 / 3) + (1 / 3)) = 1 |
18 | 10, 11, 12, 17 | subaddrii 10975 | . . . 4 ⊢ (1 − (2 / 3)) = (1 / 3) |
19 | 9, 18 | eqtri 2844 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 / 3) |
20 | 1re 10641 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | 0lt1 11162 | . . . . 5 ⊢ 0 < 1 | |
22 | 1le1 11268 | . . . . 5 ⊢ 1 ≤ 1 | |
23 | 0xr 10688 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
24 | elioc2 12800 | . . . . . . 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 15539 | . . . . . 6 ⊢ (1 ∈ (0(,]1) → ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3)))) | |
27 | 25, 26 | sylbir 237 | . . . . 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 1457 | . . . 4 ⊢ ((1 − (2 · ((1↑2) / 3))) < (cos‘1) ∧ (cos‘1) < (1 − ((1↑2) / 3))) |
29 | 28 | simpli 486 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) < (cos‘1) |
30 | 19, 29 | eqbrtrri 5089 | . 2 ⊢ (1 / 3) < (cos‘1) |
31 | 28 | simpri 488 | . . 3 ⊢ (cos‘1) < (1 − ((1↑2) / 3)) |
32 | 2 | oveq2i 7167 | . . . 4 ⊢ (1 − ((1↑2) / 3)) = (1 − (1 / 3)) |
33 | 10, 12, 11 | subadd2i 10974 | . . . . 5 ⊢ ((1 − (1 / 3)) = (2 / 3) ↔ ((2 / 3) + (1 / 3)) = 1) |
34 | 17, 33 | mpbir 233 | . . . 4 ⊢ (1 − (1 / 3)) = (2 / 3) |
35 | 32, 34 | eqtri 2844 | . . 3 ⊢ (1 − ((1↑2) / 3)) = (2 / 3) |
36 | 31, 35 | breqtri 5091 | . 2 ⊢ (cos‘1) < (2 / 3) |
37 | 30, 36 | pm3.2i 473 | 1 ⊢ ((1 / 3) < (cos‘1) ∧ (cos‘1) < (2 / 3)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 2c2 11693 3c3 11694 (,]cioc 12740 ↑cexp 13430 cosccos 15418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-ioc 12744 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-cos 15424 |
This theorem is referenced by: cos2bnd 15541 |
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