Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13550 | . . . . . . . 8 ⊢ (1↑2) = 1 | |
2 | 1 | oveq1i 7158 | . . . . . . 7 ⊢ ((1↑2) / 3) = (1 / 3) |
3 | 2 | oveq2i 7159 | . . . . . 6 ⊢ (2 · ((1↑2) / 3)) = (2 · (1 / 3)) |
4 | 2cn 11704 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
5 | 3cn 11710 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
6 | 3ne0 11735 | . . . . . . 7 ⊢ 3 ≠ 0 | |
7 | 4, 5, 6 | divreci 11377 | . . . . . 6 ⊢ (2 / 3) = (2 · (1 / 3)) |
8 | 3, 7 | eqtr4i 2845 | . . . . 5 ⊢ (2 · ((1↑2) / 3)) = (2 / 3) |
9 | 8 | oveq2i 7159 | . . . 4 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 − (2 / 3)) |
10 | ax-1cn 10587 | . . . . 5 ⊢ 1 ∈ ℂ | |
11 | 4, 5, 6 | divcli 11374 | . . . . 5 ⊢ (2 / 3) ∈ ℂ |
12 | 5, 6 | reccli 11362 | . . . . 5 ⊢ (1 / 3) ∈ ℂ |
13 | df-3 11693 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
14 | 13 | oveq1i 7158 | . . . . . 6 ⊢ (3 / 3) = ((2 + 1) / 3) |
15 | 5, 6 | dividi 11365 | . . . . . 6 ⊢ (3 / 3) = 1 |
16 | 4, 10, 5, 6 | divdiri 11389 | . . . . . 6 ⊢ ((2 + 1) / 3) = ((2 / 3) + (1 / 3)) |
17 | 14, 15, 16 | 3eqtr3ri 2851 | . . . . 5 ⊢ ((2 / 3) + (1 / 3)) = 1 |
18 | 10, 11, 12, 17 | subaddrii 10967 | . . . 4 ⊢ (1 − (2 / 3)) = (1 / 3) |
19 | 9, 18 | eqtri 2842 | . . 3 ⊢ (1 − (2 · ((1↑2) / 3))) = (1 / 3) |
20 | 1re 10633 | . . . . 5 ⊢ 1 ∈ ℝ | |
21 | 0lt1 11154 | . . . . 5 ⊢ 0 < 1 | |
22 | 1le1 11260 | . . . . 5 ⊢ 1 ≤ 1 | |
23 | 0xr 10680 | . . . . . . 7 ⊢ 0 ∈ ℝ* | |
24 | elioc2 12791 | . . . . . . 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 15531 | . . . . . 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 1455 | . . . 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 5080 | . 2 ⊢ (1 / 3) < (cos‘1) |
31 | 28 | simpri 488 | . . 3 ⊢ (cos‘1) < (1 − ((1↑2) / 3)) |
32 | 2 | oveq2i 7159 | . . . 4 ⊢ (1 − ((1↑2) / 3)) = (1 − (1 / 3)) |
33 | 10, 12, 11 | subadd2i 10966 | . . . . 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 2842 | . . 3 ⊢ (1 − ((1↑2) / 3)) = (2 / 3) |
36 | 31, 35 | breqtri 5082 | . 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 1082 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 ℝ*cxr 10666 < clt 10667 ≤ cle 10668 − cmin 10862 / cdiv 11289 2c2 11684 3c3 11685 (,]cioc 12731 ↑cexp 13421 cosccos 15410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-ioc 12735 df-ico 12736 df-fz 12885 df-fzo 13026 df-fl 13154 df-seq 13362 df-exp 13422 df-fac 13626 df-hash 13683 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-cos 15416 |
This theorem is referenced by: cos2bnd 15533 |
Copyright terms: Public domain | W3C validator |