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Mirrors > Home > ILE Home > Th. List > sincosq1lem | GIF version |
Description: Lemma for sincosq1sgn 14250. (Contributed by Paul Chapman, 24-Jan-2008.) |
Ref | Expression |
---|---|
sincosq1lem | β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β 0 < (sinβπ΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfpire 14216 | . . . . . 6 β’ (Ο / 2) β β | |
2 | ltle 8045 | . . . . . 6 β’ ((π΄ β β β§ (Ο / 2) β β) β (π΄ < (Ο / 2) β π΄ β€ (Ο / 2))) | |
3 | 1, 2 | mpan2 425 | . . . . 5 β’ (π΄ β β β (π΄ < (Ο / 2) β π΄ β€ (Ο / 2))) |
4 | pire 14210 | . . . . . . . 8 β’ Ο β β | |
5 | 4re 8996 | . . . . . . . 8 β’ 4 β β | |
6 | pigt2lt4 14208 | . . . . . . . . 9 β’ (2 < Ο β§ Ο < 4) | |
7 | 6 | simpri 113 | . . . . . . . 8 β’ Ο < 4 |
8 | 4, 5, 7 | ltleii 8060 | . . . . . . 7 β’ Ο β€ 4 |
9 | 2re 8989 | . . . . . . . . 9 β’ 2 β β | |
10 | 2pos 9010 | . . . . . . . . . 10 β’ 0 < 2 | |
11 | 9, 10 | pm3.2i 272 | . . . . . . . . 9 β’ (2 β β β§ 0 < 2) |
12 | ledivmul 8834 | . . . . . . . . 9 β’ ((Ο β β β§ 2 β β β§ (2 β β β§ 0 < 2)) β ((Ο / 2) β€ 2 β Ο β€ (2 Β· 2))) | |
13 | 4, 9, 11, 12 | mp3an 1337 | . . . . . . . 8 β’ ((Ο / 2) β€ 2 β Ο β€ (2 Β· 2)) |
14 | 2t2e4 9073 | . . . . . . . . 9 β’ (2 Β· 2) = 4 | |
15 | 14 | breq2i 4012 | . . . . . . . 8 β’ (Ο β€ (2 Β· 2) β Ο β€ 4) |
16 | 13, 15 | bitri 184 | . . . . . . 7 β’ ((Ο / 2) β€ 2 β Ο β€ 4) |
17 | 8, 16 | mpbir 146 | . . . . . 6 β’ (Ο / 2) β€ 2 |
18 | letr 8040 | . . . . . . 7 β’ ((π΄ β β β§ (Ο / 2) β β β§ 2 β β) β ((π΄ β€ (Ο / 2) β§ (Ο / 2) β€ 2) β π΄ β€ 2)) | |
19 | 1, 9, 18 | mp3an23 1329 | . . . . . 6 β’ (π΄ β β β ((π΄ β€ (Ο / 2) β§ (Ο / 2) β€ 2) β π΄ β€ 2)) |
20 | 17, 19 | mpan2i 431 | . . . . 5 β’ (π΄ β β β (π΄ β€ (Ο / 2) β π΄ β€ 2)) |
21 | 3, 20 | syld 45 | . . . 4 β’ (π΄ β β β (π΄ < (Ο / 2) β π΄ β€ 2)) |
22 | 21 | adantr 276 | . . 3 β’ ((π΄ β β β§ 0 < π΄) β (π΄ < (Ο / 2) β π΄ β€ 2)) |
23 | 22 | 3impia 1200 | . 2 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β π΄ β€ 2) |
24 | 0xr 8004 | . . . 4 β’ 0 β β* | |
25 | elioc2 9936 | . . . 4 β’ ((0 β β* β§ 2 β β) β (π΄ β (0(,]2) β (π΄ β β β§ 0 < π΄ β§ π΄ β€ 2))) | |
26 | 24, 9, 25 | mp2an 426 | . . 3 β’ (π΄ β (0(,]2) β (π΄ β β β§ 0 < π΄ β§ π΄ β€ 2)) |
27 | sin02gt0 11771 | . . 3 β’ (π΄ β (0(,]2) β 0 < (sinβπ΄)) | |
28 | 26, 27 | sylbir 135 | . 2 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ β€ 2) β 0 < (sinβπ΄)) |
29 | 23, 28 | syld3an3 1283 | 1 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ < (Ο / 2)) β 0 < (sinβπ΄)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 β§ w3a 978 β wcel 2148 class class class wbr 4004 βcfv 5217 (class class class)co 5875 βcr 7810 0cc0 7811 Β· cmul 7816 β*cxr 7991 < clt 7992 β€ cle 7993 / cdiv 8629 2c2 8970 4c4 8972 (,]cioc 9889 sincsin 11652 Οcpi 11655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929 ax-arch 7930 ax-caucvg 7931 ax-pre-suploc 7932 ax-addf 7933 ax-mulf 7934 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-disj 3982 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-isom 5226 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-of 6083 df-1st 6141 df-2nd 6142 df-recs 6306 df-irdg 6371 df-frec 6392 df-1o 6417 df-oadd 6421 df-er 6535 df-map 6650 df-pm 6651 df-en 6741 df-dom 6742 df-fin 6743 df-sup 6983 df-inf 6984 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-5 8981 df-6 8982 df-7 8983 df-8 8984 df-9 8985 df-n0 9177 df-z 9254 df-uz 9529 df-q 9620 df-rp 9654 df-xneg 9772 df-xadd 9773 df-ioo 9892 df-ioc 9893 df-ico 9894 df-icc 9895 df-fz 10009 df-fzo 10143 df-seqfrec 10446 df-exp 10520 df-fac 10706 df-bc 10728 df-ihash 10756 df-shft 10824 df-cj 10851 df-re 10852 df-im 10853 df-rsqrt 11007 df-abs 11008 df-clim 11287 df-sumdc 11362 df-ef 11656 df-sin 11658 df-cos 11659 df-pi 11661 df-rest 12690 df-topgen 12709 df-psmet 13450 df-xmet 13451 df-met 13452 df-bl 13453 df-mopn 13454 df-top 13501 df-topon 13514 df-bases 13546 df-ntr 13599 df-cn 13691 df-cnp 13692 df-tx 13756 df-cncf 14061 df-limced 14128 df-dvap 14129 |
This theorem is referenced by: sincosq1sgn 14250 sinq12gt0 14254 |
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