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Mirrors > Home > ILE Home > Th. List > cosbnd | GIF version |
Description: The cosine of a real number lies between -1 and 1. Equation 18 of [Gleason] p. 311. (Contributed by NM, 16-Jan-2006.) |
Ref | Expression |
---|---|
cosbnd | β’ (π΄ β β β (-1 β€ (cosβπ΄) β§ (cosβπ΄) β€ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resincl 11730 | . . . . . 6 β’ (π΄ β β β (sinβπ΄) β β) | |
2 | 1 | sqge0d 10683 | . . . . 5 β’ (π΄ β β β 0 β€ ((sinβπ΄)β2)) |
3 | recoscl 11731 | . . . . . . 7 β’ (π΄ β β β (cosβπ΄) β β) | |
4 | 3 | resqcld 10682 | . . . . . 6 β’ (π΄ β β β ((cosβπ΄)β2) β β) |
5 | 1 | resqcld 10682 | . . . . . 6 β’ (π΄ β β β ((sinβπ΄)β2) β β) |
6 | 4, 5 | addge02d 8493 | . . . . 5 β’ (π΄ β β β (0 β€ ((sinβπ΄)β2) β ((cosβπ΄)β2) β€ (((sinβπ΄)β2) + ((cosβπ΄)β2)))) |
7 | 2, 6 | mpbid 147 | . . . 4 β’ (π΄ β β β ((cosβπ΄)β2) β€ (((sinβπ΄)β2) + ((cosβπ΄)β2))) |
8 | recn 7946 | . . . . . 6 β’ (π΄ β β β π΄ β β) | |
9 | sincossq 11758 | . . . . . 6 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) | |
10 | 8, 9 | syl 14 | . . . . 5 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = 1) |
11 | sq1 10616 | . . . . 5 β’ (1β2) = 1 | |
12 | 10, 11 | eqtr4di 2228 | . . . 4 β’ (π΄ β β β (((sinβπ΄)β2) + ((cosβπ΄)β2)) = (1β2)) |
13 | 7, 12 | breqtrd 4031 | . . 3 β’ (π΄ β β β ((cosβπ΄)β2) β€ (1β2)) |
14 | 1re 7958 | . . . . . 6 β’ 1 β β | |
15 | 0le1 8440 | . . . . . 6 β’ 0 β€ 1 | |
16 | lenegsq 11106 | . . . . . 6 β’ (((cosβπ΄) β β β§ 1 β β β§ 0 β€ 1) β (((cosβπ΄) β€ 1 β§ -(cosβπ΄) β€ 1) β ((cosβπ΄)β2) β€ (1β2))) | |
17 | 14, 15, 16 | mp3an23 1329 | . . . . 5 β’ ((cosβπ΄) β β β (((cosβπ΄) β€ 1 β§ -(cosβπ΄) β€ 1) β ((cosβπ΄)β2) β€ (1β2))) |
18 | lenegcon1 8425 | . . . . . . 7 β’ (((cosβπ΄) β β β§ 1 β β) β (-(cosβπ΄) β€ 1 β -1 β€ (cosβπ΄))) | |
19 | 14, 18 | mpan2 425 | . . . . . 6 β’ ((cosβπ΄) β β β (-(cosβπ΄) β€ 1 β -1 β€ (cosβπ΄))) |
20 | 19 | anbi2d 464 | . . . . 5 β’ ((cosβπ΄) β β β (((cosβπ΄) β€ 1 β§ -(cosβπ΄) β€ 1) β ((cosβπ΄) β€ 1 β§ -1 β€ (cosβπ΄)))) |
21 | 17, 20 | bitr3d 190 | . . . 4 β’ ((cosβπ΄) β β β (((cosβπ΄)β2) β€ (1β2) β ((cosβπ΄) β€ 1 β§ -1 β€ (cosβπ΄)))) |
22 | 3, 21 | syl 14 | . . 3 β’ (π΄ β β β (((cosβπ΄)β2) β€ (1β2) β ((cosβπ΄) β€ 1 β§ -1 β€ (cosβπ΄)))) |
23 | 13, 22 | mpbid 147 | . 2 β’ (π΄ β β β ((cosβπ΄) β€ 1 β§ -1 β€ (cosβπ΄))) |
24 | 23 | ancomd 267 | 1 β’ (π΄ β β β (-1 β€ (cosβπ΄) β§ (cosβπ΄) β€ 1)) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wb 105 = wceq 1353 β wcel 2148 class class class wbr 4005 βcfv 5218 (class class class)co 5877 βcc 7811 βcr 7812 0cc0 7813 1c1 7814 + caddc 7816 β€ cle 7995 -cneg 8131 2c2 8972 βcexp 10521 sincsin 11654 cosccos 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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 ax-arch 7932 ax-caucvg 7933 |
This theorem depends on definitions: df-bi 117 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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-disj 3983 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-isom 5227 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-sup 6985 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-n0 9179 df-z 9256 df-uz 9531 df-q 9622 df-rp 9656 df-ico 9896 df-fz 10011 df-fzo 10145 df-seqfrec 10448 df-exp 10522 df-fac 10708 df-bc 10730 df-ihash 10758 df-cj 10853 df-re 10854 df-im 10855 df-rsqrt 11009 df-abs 11010 df-clim 11289 df-sumdc 11364 df-ef 11658 df-sin 11660 df-cos 11661 |
This theorem is referenced by: cosbnd2 11765 |
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