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Mirrors > Home > MPE Home > Th. List > sincosq1sgn | Structured version Visualization version GIF version |
Description: The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.) |
Ref | Expression |
---|---|
sincosq1sgn | β’ (π΄ β (0(,)(Ο / 2)) β (0 < (sinβπ΄) β§ 0 < (cosβπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 11209 | . . 3 β’ 0 β β* | |
2 | halfpire 25837 | . . . 4 β’ (Ο / 2) β β | |
3 | 2 | rexri 11220 | . . 3 β’ (Ο / 2) β β* |
4 | elioo2 13312 | . . 3 β’ ((0 β β* β§ (Ο / 2) β β*) β (π΄ β (0(,)(Ο / 2)) β (π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)))) | |
5 | 1, 3, 4 | mp2an 691 | . 2 β’ (π΄ β (0(,)(Ο / 2)) β (π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2))) |
6 | sincosq1lem 25870 | . . 3 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β 0 < (sinβπ΄)) | |
7 | resubcl 11472 | . . . . . . . 8 β’ (((Ο / 2) β β β§ π΄ β β) β ((Ο / 2) β π΄) β β) | |
8 | 2, 7 | mpan 689 | . . . . . . 7 β’ (π΄ β β β ((Ο / 2) β π΄) β β) |
9 | sincosq1lem 25870 | . . . . . . 7 β’ ((((Ο / 2) β π΄) β β β§ 0 < ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) < (Ο / 2)) β 0 < (sinβ((Ο / 2) β π΄))) | |
10 | 8, 9 | syl3an1 1164 | . . . . . 6 β’ ((π΄ β β β§ 0 < ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) < (Ο / 2)) β 0 < (sinβ((Ο / 2) β π΄))) |
11 | 10 | 3expib 1123 | . . . . 5 β’ (π΄ β β β ((0 < ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) < (Ο / 2)) β 0 < (sinβ((Ο / 2) β π΄)))) |
12 | 0re 11164 | . . . . . . . . 9 β’ 0 β β | |
13 | ltsub13 11643 | . . . . . . . . 9 β’ ((0 β β β§ (Ο / 2) β β β§ π΄ β β) β (0 < ((Ο / 2) β π΄) β π΄ < ((Ο / 2) β 0))) | |
14 | 12, 2, 13 | mp3an12 1452 | . . . . . . . 8 β’ (π΄ β β β (0 < ((Ο / 2) β π΄) β π΄ < ((Ο / 2) β 0))) |
15 | 2 | recni 11176 | . . . . . . . . . 10 β’ (Ο / 2) β β |
16 | 15 | subid1i 11480 | . . . . . . . . 9 β’ ((Ο / 2) β 0) = (Ο / 2) |
17 | 16 | breq2i 5118 | . . . . . . . 8 β’ (π΄ < ((Ο / 2) β 0) β π΄ < (Ο / 2)) |
18 | 14, 17 | bitrdi 287 | . . . . . . 7 β’ (π΄ β β β (0 < ((Ο / 2) β π΄) β π΄ < (Ο / 2))) |
19 | ltsub23 11642 | . . . . . . . . 9 β’ (((Ο / 2) β β β§ π΄ β β β§ (Ο / 2) β β) β (((Ο / 2) β π΄) < (Ο / 2) β ((Ο / 2) β (Ο / 2)) < π΄)) | |
20 | 2, 2, 19 | mp3an13 1453 | . . . . . . . 8 β’ (π΄ β β β (((Ο / 2) β π΄) < (Ο / 2) β ((Ο / 2) β (Ο / 2)) < π΄)) |
21 | 15 | subidi 11479 | . . . . . . . . 9 β’ ((Ο / 2) β (Ο / 2)) = 0 |
22 | 21 | breq1i 5117 | . . . . . . . 8 β’ (((Ο / 2) β (Ο / 2)) < π΄ β 0 < π΄) |
23 | 20, 22 | bitrdi 287 | . . . . . . 7 β’ (π΄ β β β (((Ο / 2) β π΄) < (Ο / 2) β 0 < π΄)) |
24 | 18, 23 | anbi12d 632 | . . . . . 6 β’ (π΄ β β β ((0 < ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) < (Ο / 2)) β (π΄ < (Ο / 2) β§ 0 < π΄))) |
25 | 24 | biancomd 465 | . . . . 5 β’ (π΄ β β β ((0 < ((Ο / 2) β π΄) β§ ((Ο / 2) β π΄) < (Ο / 2)) β (0 < π΄ β§ π΄ < (Ο / 2)))) |
26 | recn 11148 | . . . . . . 7 β’ (π΄ β β β π΄ β β) | |
27 | sinhalfpim 25866 | . . . . . . 7 β’ (π΄ β β β (sinβ((Ο / 2) β π΄)) = (cosβπ΄)) | |
28 | 26, 27 | syl 17 | . . . . . 6 β’ (π΄ β β β (sinβ((Ο / 2) β π΄)) = (cosβπ΄)) |
29 | 28 | breq2d 5122 | . . . . 5 β’ (π΄ β β β (0 < (sinβ((Ο / 2) β π΄)) β 0 < (cosβπ΄))) |
30 | 11, 25, 29 | 3imtr3d 293 | . . . 4 β’ (π΄ β β β ((0 < π΄ β§ π΄ < (Ο / 2)) β 0 < (cosβπ΄))) |
31 | 30 | 3impib 1117 | . . 3 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β 0 < (cosβπ΄)) |
32 | 6, 31 | jca 513 | . 2 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β (0 < (sinβπ΄) β§ 0 < (cosβπ΄))) |
33 | 5, 32 | sylbi 216 | 1 β’ (π΄ β (0(,)(Ο / 2)) β (0 < (sinβπ΄) β§ 0 < (cosβπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 class class class wbr 5110 βcfv 6501 (class class class)co 7362 βcc 11056 βcr 11057 0cc0 11058 β*cxr 11195 < clt 11196 β cmin 11392 / cdiv 11819 2c2 12215 (,)cioo 13271 sincsin 15953 cosccos 15954 Οcpi 15956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14959 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-ef 15957 df-sin 15959 df-cos 15960 df-pi 15962 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lp 22503 df-perf 22504 df-cn 22594 df-cnp 22595 df-haus 22682 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-limc 25246 df-dv 25247 |
This theorem is referenced by: sincosq2sgn 25872 coseq00topi 25875 tanrpcl 25877 tangtx 25878 tanabsge 25879 sincos6thpi 25888 tanord1 25909 basellem3 26448 basellem4 26449 basellem8 26453 |
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