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Mirrors > Home > MPE Home > Th. List > sincos2sgn | Structured version Visualization version GIF version |
Description: The signs of the sine and cosine of 2. (Contributed by Paul Chapman, 19-Jan-2008.) |
Ref | Expression |
---|---|
sincos2sgn | ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2re 11432 | . . . 4 ⊢ 2 ∈ ℝ | |
2 | 2pos 11468 | . . . 4 ⊢ 0 < 2 | |
3 | 1 | leidi 10893 | . . . 4 ⊢ 2 ≤ 2 |
4 | 0xr 10410 | . . . . 5 ⊢ 0 ∈ ℝ* | |
5 | elioc2 12531 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 2 ∈ ℝ) → (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2))) | |
6 | 4, 1, 5 | mp2an 683 | . . . 4 ⊢ (2 ∈ (0(,]2) ↔ (2 ∈ ℝ ∧ 0 < 2 ∧ 2 ≤ 2)) |
7 | 1, 2, 3, 6 | mpbir3an 1445 | . . 3 ⊢ 2 ∈ (0(,]2) |
8 | sin02gt0 15301 | . . 3 ⊢ (2 ∈ (0(,]2) → 0 < (sin‘2)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ 0 < (sin‘2) |
10 | cos2bnd 15297 | . . . 4 ⊢ (-(7 / 9) < (cos‘2) ∧ (cos‘2) < -(1 / 9)) | |
11 | 10 | simpri 481 | . . 3 ⊢ (cos‘2) < -(1 / 9) |
12 | 9re 11463 | . . . . 5 ⊢ 9 ∈ ℝ | |
13 | 9pos 11478 | . . . . 5 ⊢ 0 < 9 | |
14 | 12, 13 | recgt0ii 11266 | . . . 4 ⊢ 0 < (1 / 9) |
15 | 12, 13 | gt0ne0ii 10895 | . . . . . 6 ⊢ 9 ≠ 0 |
16 | 12, 15 | rereccli 11123 | . . . . 5 ⊢ (1 / 9) ∈ ℝ |
17 | lt0neg2 10866 | . . . . 5 ⊢ ((1 / 9) ∈ ℝ → (0 < (1 / 9) ↔ -(1 / 9) < 0)) | |
18 | 16, 17 | ax-mp 5 | . . . 4 ⊢ (0 < (1 / 9) ↔ -(1 / 9) < 0) |
19 | 14, 18 | mpbi 222 | . . 3 ⊢ -(1 / 9) < 0 |
20 | recoscl 15250 | . . . . 5 ⊢ (2 ∈ ℝ → (cos‘2) ∈ ℝ) | |
21 | 1, 20 | ax-mp 5 | . . . 4 ⊢ (cos‘2) ∈ ℝ |
22 | 16 | renegcli 10670 | . . . 4 ⊢ -(1 / 9) ∈ ℝ |
23 | 0re 10365 | . . . 4 ⊢ 0 ∈ ℝ | |
24 | 21, 22, 23 | lttri 10489 | . . 3 ⊢ (((cos‘2) < -(1 / 9) ∧ -(1 / 9) < 0) → (cos‘2) < 0) |
25 | 11, 19, 24 | mp2an 683 | . 2 ⊢ (cos‘2) < 0 |
26 | 9, 25 | pm3.2i 464 | 1 ⊢ (0 < (sin‘2) ∧ (cos‘2) < 0) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∧ w3a 1111 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 0cc0 10259 1c1 10260 ℝ*cxr 10397 < clt 10398 ≤ cle 10399 -cneg 10593 / cdiv 11016 2c2 11413 7c7 11418 9c9 11420 (,]cioc 12471 sincsin 15173 cosccos 15174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-ioc 12475 df-ico 12476 df-fz 12627 df-fzo 12768 df-fl 12895 df-seq 13103 df-exp 13162 df-fac 13361 df-bc 13390 df-hash 13418 df-shft 14191 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-limsup 14586 df-clim 14603 df-rlim 14604 df-sum 14801 df-ef 15177 df-sin 15179 df-cos 15180 |
This theorem is referenced by: sin4lt0 15304 pilem3 24613 pilem3OLD 24614 |
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