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Mirrors > Home > MPE Home > Th. List > cosq14ge0 | Structured version Visualization version GIF version |
Description: The cosine of a number between -Ο / 2 and Ο / 2 is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.) |
Ref | Expression |
---|---|
cosq14ge0 | β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β 0 β€ (cosβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfpire 26315 | . . . . 5 β’ (Ο / 2) β β | |
2 | neghalfpire 26316 | . . . . . . 7 β’ -(Ο / 2) β β | |
3 | 2, 1 | elicc2i 13386 | . . . . . 6 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (π΄ β β β§ -(Ο / 2) β€ π΄ β§ π΄ β€ (Ο / 2))) |
4 | 3 | simp1bi 1142 | . . . . 5 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β π΄ β β) |
5 | resubcl 11520 | . . . . 5 β’ (((Ο / 2) β β β§ π΄ β β) β ((Ο / 2) β π΄) β β) | |
6 | 1, 4, 5 | sylancr 586 | . . . 4 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β ((Ο / 2) β π΄) β β) |
7 | 3 | simp3bi 1144 | . . . . 5 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β π΄ β€ (Ο / 2)) |
8 | subge0 11723 | . . . . . 6 β’ (((Ο / 2) β β β§ π΄ β β) β (0 β€ ((Ο / 2) β π΄) β π΄ β€ (Ο / 2))) | |
9 | 1, 4, 8 | sylancr 586 | . . . . 5 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (0 β€ ((Ο / 2) β π΄) β π΄ β€ (Ο / 2))) |
10 | 7, 9 | mpbird 257 | . . . 4 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β 0 β€ ((Ο / 2) β π΄)) |
11 | picn 26310 | . . . . . . . 8 β’ Ο β β | |
12 | halfcl 12433 | . . . . . . . 8 β’ (Ο β β β (Ο / 2) β β) | |
13 | 11, 12 | ax-mp 5 | . . . . . . 7 β’ (Ο / 2) β β |
14 | 13 | negcli 11524 | . . . . . . 7 β’ -(Ο / 2) β β |
15 | 11, 13 | negsubi 11534 | . . . . . . . 8 β’ (Ο + -(Ο / 2)) = (Ο β (Ο / 2)) |
16 | pidiv2halves 26318 | . . . . . . . . 9 β’ ((Ο / 2) + (Ο / 2)) = Ο | |
17 | 11, 13, 13, 16 | subaddrii 11545 | . . . . . . . 8 β’ (Ο β (Ο / 2)) = (Ο / 2) |
18 | 15, 17 | eqtri 2752 | . . . . . . 7 β’ (Ο + -(Ο / 2)) = (Ο / 2) |
19 | 13, 11, 14, 18 | subaddrii 11545 | . . . . . 6 β’ ((Ο / 2) β Ο) = -(Ο / 2) |
20 | 3 | simp2bi 1143 | . . . . . 6 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β -(Ο / 2) β€ π΄) |
21 | 19, 20 | eqbrtrid 5173 | . . . . 5 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β ((Ο / 2) β Ο) β€ π΄) |
22 | pire 26309 | . . . . . . 7 β’ Ο β β | |
23 | suble 11688 | . . . . . . 7 β’ (((Ο / 2) β β β§ π΄ β β β§ Ο β β) β (((Ο / 2) β π΄) β€ Ο β ((Ο / 2) β Ο) β€ π΄)) | |
24 | 1, 22, 23 | mp3an13 1448 | . . . . . 6 β’ (π΄ β β β (((Ο / 2) β π΄) β€ Ο β ((Ο / 2) β Ο) β€ π΄)) |
25 | 4, 24 | syl 17 | . . . . 5 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (((Ο / 2) β π΄) β€ Ο β ((Ο / 2) β Ο) β€ π΄)) |
26 | 21, 25 | mpbird 257 | . . . 4 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β ((Ο / 2) β π΄) β€ Ο) |
27 | 0re 11212 | . . . . 5 β’ 0 β β | |
28 | 27, 22 | elicc2i 13386 | . . . 4 β’ (((Ο / 2) β π΄) β (0[,]Ο) β (((Ο / 2) β π΄) β β β§ 0 β€ ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) β€ Ο)) |
29 | 6, 10, 26, 28 | syl3anbrc 1340 | . . 3 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β ((Ο / 2) β π΄) β (0[,]Ο)) |
30 | sinq12ge0 26359 | . . 3 β’ (((Ο / 2) β π΄) β (0[,]Ο) β 0 β€ (sinβ((Ο / 2) β π΄))) | |
31 | 29, 30 | syl 17 | . 2 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β 0 β€ (sinβ((Ο / 2) β π΄))) |
32 | 4 | recnd 11238 | . . 3 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β π΄ β β) |
33 | sinhalfpim 26344 | . . 3 β’ (π΄ β β β (sinβ((Ο / 2) β π΄)) = (cosβπ΄)) | |
34 | 32, 33 | syl 17 | . 2 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β (sinβ((Ο / 2) β π΄)) = (cosβπ΄)) |
35 | 31, 34 | breqtrd 5164 | 1 β’ (π΄ β (-(Ο / 2)[,](Ο / 2)) β 0 β€ (cosβπ΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 class class class wbr 5138 βcfv 6533 (class class class)co 7401 βcc 11103 βcr 11104 0cc0 11105 + caddc 11108 β€ cle 11245 β cmin 11440 -cneg 11441 / cdiv 11867 2c2 12263 [,]cicc 13323 sincsin 16003 cosccos 16004 Οcpi 16006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-starv 17210 df-sca 17211 df-vsca 17212 df-ip 17213 df-tset 17214 df-ple 17215 df-ds 17217 df-unif 17218 df-hom 17219 df-cco 17220 df-rest 17366 df-topn 17367 df-0g 17385 df-gsum 17386 df-topgen 17387 df-pt 17388 df-prds 17391 df-xrs 17446 df-qtop 17451 df-imas 17452 df-xps 17454 df-mre 17528 df-mrc 17529 df-acs 17531 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-mulg 18985 df-cntz 19222 df-cmn 19691 df-psmet 21219 df-xmet 21220 df-met 21221 df-bl 21222 df-mopn 21223 df-fbas 21224 df-fg 21225 df-cnfld 21228 df-top 22717 df-topon 22734 df-topsp 22756 df-bases 22770 df-cld 22844 df-ntr 22845 df-cls 22846 df-nei 22923 df-lp 22961 df-perf 22962 df-cn 23052 df-cnp 23053 df-haus 23140 df-tx 23387 df-hmeo 23580 df-fil 23671 df-fm 23763 df-flim 23764 df-flf 23765 df-xms 24147 df-ms 24148 df-tms 24149 df-cncf 24719 df-limc 25716 df-dv 25717 |
This theorem is referenced by: efif1olem4 26395 cxpsqrtlem 26551 cos2h 36935 |
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