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Theorem cosordlem 12946
Description: Cosine is decreasing over the closed interval from  0 to  pi. (Contributed by Mario Carneiro, 10-May-2014.)
Hypotheses
Ref Expression
cosord.1  |-  ( ph  ->  A  e.  ( 0 [,] pi ) )
cosord.2  |-  ( ph  ->  B  e.  ( 0 [,] pi ) )
cosord.3  |-  ( ph  ->  A  <  B )
Assertion
Ref Expression
cosordlem  |-  ( ph  ->  ( cos `  B
)  <  ( cos `  A ) )

Proof of Theorem cosordlem
StepHypRef Expression
1 cosord.2 . . . . . . 7  |-  ( ph  ->  B  e.  ( 0 [,] pi ) )
2 0re 7773 . . . . . . . 8  |-  0  e.  RR
3 pire 12883 . . . . . . . 8  |-  pi  e.  RR
42, 3elicc2i 9729 . . . . . . 7  |-  ( B  e.  ( 0 [,] pi )  <->  ( B  e.  RR  /\  0  <_  B  /\  B  <_  pi ) )
51, 4sylib 121 . . . . . 6  |-  ( ph  ->  ( B  e.  RR  /\  0  <_  B  /\  B  <_  pi ) )
65simp1d 993 . . . . 5  |-  ( ph  ->  B  e.  RR )
76recnd 7801 . . . 4  |-  ( ph  ->  B  e.  CC )
8 cosord.1 . . . . . . 7  |-  ( ph  ->  A  e.  ( 0 [,] pi ) )
92, 3elicc2i 9729 . . . . . . 7  |-  ( A  e.  ( 0 [,] pi )  <->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
108, 9sylib 121 . . . . . 6  |-  ( ph  ->  ( A  e.  RR  /\  0  <_  A  /\  A  <_  pi ) )
1110simp1d 993 . . . . 5  |-  ( ph  ->  A  e.  RR )
1211recnd 7801 . . . 4  |-  ( ph  ->  A  e.  CC )
13 subcos 11460 . . . 4  |-  ( ( B  e.  CC  /\  A  e.  CC )  ->  ( ( cos `  A
)  -  ( cos `  B ) )  =  ( 2  x.  (
( sin `  (
( B  +  A
)  /  2 ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) ) ) )
147, 12, 13syl2anc 408 . . 3  |-  ( ph  ->  ( ( cos `  A
)  -  ( cos `  B ) )  =  ( 2  x.  (
( sin `  (
( B  +  A
)  /  2 ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) ) ) )
15 2rp 9453 . . . 4  |-  2  e.  RR+
166, 11readdcld 7802 . . . . . . . 8  |-  ( ph  ->  ( B  +  A
)  e.  RR )
1716rehalfcld 8973 . . . . . . 7  |-  ( ph  ->  ( ( B  +  A )  /  2
)  e.  RR )
1817resincld 11436 . . . . . 6  |-  ( ph  ->  ( sin `  (
( B  +  A
)  /  2 ) )  e.  RR )
192a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  0  e.  RR )
2010simp2d 994 . . . . . . . . . . 11  |-  ( ph  ->  0  <_  A )
21 cosord.3 . . . . . . . . . . 11  |-  ( ph  ->  A  <  B )
2219, 11, 6, 20, 21lelttrd 7894 . . . . . . . . . 10  |-  ( ph  ->  0  <  B )
236, 11, 22, 20addgtge0d 8289 . . . . . . . . 9  |-  ( ph  ->  0  <  ( B  +  A ) )
24 2re 8797 . . . . . . . . . 10  |-  2  e.  RR
25 2pos 8818 . . . . . . . . . 10  |-  0  <  2
26 divgt0 8637 . . . . . . . . . 10  |-  ( ( ( ( B  +  A )  e.  RR  /\  0  <  ( B  +  A ) )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( ( B  +  A )  /  2 ) )
2724, 25, 26mpanr12 435 . . . . . . . . 9  |-  ( ( ( B  +  A
)  e.  RR  /\  0  <  ( B  +  A ) )  -> 
0  <  ( ( B  +  A )  /  2 ) )
2816, 23, 27syl2anc 408 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( B  +  A )  /  2 ) )
293a1i 9 . . . . . . . . 9  |-  ( ph  ->  pi  e.  RR )
3011, 6, 6, 21ltadd2dd 8191 . . . . . . . . . . 11  |-  ( ph  ->  ( B  +  A
)  <  ( B  +  B ) )
3172timesd 8969 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  B
)  =  ( B  +  B ) )
3230, 31breqtrrd 3956 . . . . . . . . . 10  |-  ( ph  ->  ( B  +  A
)  <  ( 2  x.  B ) )
3324a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  RR )
3425a1i 9 . . . . . . . . . . 11  |-  ( ph  ->  0  <  2 )
35 ltdivmul 8641 . . . . . . . . . . 11  |-  ( ( ( B  +  A
)  e.  RR  /\  B  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( ( B  +  A )  /  2 )  < 
B  <->  ( B  +  A )  <  (
2  x.  B ) ) )
3616, 6, 33, 34, 35syl112anc 1220 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( B  +  A )  / 
2 )  <  B  <->  ( B  +  A )  <  ( 2  x.  B ) ) )
3732, 36mpbird 166 . . . . . . . . 9  |-  ( ph  ->  ( ( B  +  A )  /  2
)  <  B )
385simp3d 995 . . . . . . . . 9  |-  ( ph  ->  B  <_  pi )
3917, 6, 29, 37, 38ltletrd 8192 . . . . . . . 8  |-  ( ph  ->  ( ( B  +  A )  /  2
)  <  pi )
40 0xr 7819 . . . . . . . . 9  |-  0  e.  RR*
413rexri 7830 . . . . . . . . 9  |-  pi  e.  RR*
42 elioo2 9711 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  (
( ( B  +  A )  /  2
)  e.  ( 0 (,) pi )  <->  ( (
( B  +  A
)  /  2 )  e.  RR  /\  0  <  ( ( B  +  A )  /  2
)  /\  ( ( B  +  A )  /  2 )  < 
pi ) ) )
4340, 41, 42mp2an 422 . . . . . . . 8  |-  ( ( ( B  +  A
)  /  2 )  e.  ( 0 (,) pi )  <->  ( (
( B  +  A
)  /  2 )  e.  RR  /\  0  <  ( ( B  +  A )  /  2
)  /\  ( ( B  +  A )  /  2 )  < 
pi ) )
4417, 28, 39, 43syl3anbrc 1165 . . . . . . 7  |-  ( ph  ->  ( ( B  +  A )  /  2
)  e.  ( 0 (,) pi ) )
45 sinq12gt0 12927 . . . . . . 7  |-  ( ( ( B  +  A
)  /  2 )  e.  ( 0 (,) pi )  ->  0  <  ( sin `  (
( B  +  A
)  /  2 ) ) )
4644, 45syl 14 . . . . . 6  |-  ( ph  ->  0  <  ( sin `  ( ( B  +  A )  /  2
) ) )
4718, 46elrpd 9488 . . . . 5  |-  ( ph  ->  ( sin `  (
( B  +  A
)  /  2 ) )  e.  RR+ )
486, 11resubcld 8150 . . . . . . . 8  |-  ( ph  ->  ( B  -  A
)  e.  RR )
4948rehalfcld 8973 . . . . . . 7  |-  ( ph  ->  ( ( B  -  A )  /  2
)  e.  RR )
5049resincld 11436 . . . . . 6  |-  ( ph  ->  ( sin `  (
( B  -  A
)  /  2 ) )  e.  RR )
5111, 6posdifd 8301 . . . . . . . . . 10  |-  ( ph  ->  ( A  <  B  <->  0  <  ( B  -  A ) ) )
5221, 51mpbid 146 . . . . . . . . 9  |-  ( ph  ->  0  <  ( B  -  A ) )
53 divgt0 8637 . . . . . . . . . 10  |-  ( ( ( ( B  -  A )  e.  RR  /\  0  <  ( B  -  A ) )  /\  ( 2  e.  RR  /\  0  <  2 ) )  -> 
0  <  ( ( B  -  A )  /  2 ) )
5424, 25, 53mpanr12 435 . . . . . . . . 9  |-  ( ( ( B  -  A
)  e.  RR  /\  0  <  ( B  -  A ) )  -> 
0  <  ( ( B  -  A )  /  2 ) )
5548, 52, 54syl2anc 408 . . . . . . . 8  |-  ( ph  ->  0  <  ( ( B  -  A )  /  2 ) )
56 rehalfcl 8954 . . . . . . . . . 10  |-  ( pi  e.  RR  ->  (
pi  /  2 )  e.  RR )
573, 56mp1i 10 . . . . . . . . 9  |-  ( ph  ->  ( pi  /  2
)  e.  RR )
586, 11subge02d 8306 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  A  <->  ( B  -  A )  <_  B ) )
5920, 58mpbid 146 . . . . . . . . . . 11  |-  ( ph  ->  ( B  -  A
)  <_  B )
6048, 6, 29, 59, 38letrd 7893 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  A
)  <_  pi )
61 lediv1 8634 . . . . . . . . . . 11  |-  ( ( ( B  -  A
)  e.  RR  /\  pi  e.  RR  /\  (
2  e.  RR  /\  0  <  2 ) )  ->  ( ( B  -  A )  <_  pi 
<->  ( ( B  -  A )  /  2
)  <_  ( pi  /  2 ) ) )
6248, 29, 33, 34, 61syl112anc 1220 . . . . . . . . . 10  |-  ( ph  ->  ( ( B  -  A )  <_  pi  <->  ( ( B  -  A
)  /  2 )  <_  ( pi  / 
2 ) ) )
6360, 62mpbid 146 . . . . . . . . 9  |-  ( ph  ->  ( ( B  -  A )  /  2
)  <_  ( pi  /  2 ) )
64 pirp 12886 . . . . . . . . . 10  |-  pi  e.  RR+
65 rphalflt 9478 . . . . . . . . . 10  |-  ( pi  e.  RR+  ->  ( pi 
/  2 )  < 
pi )
6664, 65mp1i 10 . . . . . . . . 9  |-  ( ph  ->  ( pi  /  2
)  <  pi )
6749, 57, 29, 63, 66lelttrd 7894 . . . . . . . 8  |-  ( ph  ->  ( ( B  -  A )  /  2
)  <  pi )
68 elioo2 9711 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  pi  e.  RR* )  ->  (
( ( B  -  A )  /  2
)  e.  ( 0 (,) pi )  <->  ( (
( B  -  A
)  /  2 )  e.  RR  /\  0  <  ( ( B  -  A )  /  2
)  /\  ( ( B  -  A )  /  2 )  < 
pi ) ) )
6940, 41, 68mp2an 422 . . . . . . . 8  |-  ( ( ( B  -  A
)  /  2 )  e.  ( 0 (,) pi )  <->  ( (
( B  -  A
)  /  2 )  e.  RR  /\  0  <  ( ( B  -  A )  /  2
)  /\  ( ( B  -  A )  /  2 )  < 
pi ) )
7049, 55, 67, 69syl3anbrc 1165 . . . . . . 7  |-  ( ph  ->  ( ( B  -  A )  /  2
)  e.  ( 0 (,) pi ) )
71 sinq12gt0 12927 . . . . . . 7  |-  ( ( ( B  -  A
)  /  2 )  e.  ( 0 (,) pi )  ->  0  <  ( sin `  (
( B  -  A
)  /  2 ) ) )
7270, 71syl 14 . . . . . 6  |-  ( ph  ->  0  <  ( sin `  ( ( B  -  A )  /  2
) ) )
7350, 72elrpd 9488 . . . . 5  |-  ( ph  ->  ( sin `  (
( B  -  A
)  /  2 ) )  e.  RR+ )
7447, 73rpmulcld 9507 . . . 4  |-  ( ph  ->  ( ( sin `  (
( B  +  A
)  /  2 ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  e.  RR+ )
75 rpmulcl 9473 . . . 4  |-  ( ( 2  e.  RR+  /\  (
( sin `  (
( B  +  A
)  /  2 ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) )  e.  RR+ )  ->  ( 2  x.  ( ( sin `  ( ( B  +  A )  /  2
) )  x.  ( sin `  ( ( B  -  A )  / 
2 ) ) ) )  e.  RR+ )
7615, 74, 75sylancr 410 . . 3  |-  ( ph  ->  ( 2  x.  (
( sin `  (
( B  +  A
)  /  2 ) )  x.  ( sin `  ( ( B  -  A )  /  2
) ) ) )  e.  RR+ )
7714, 76eqeltrd 2216 . 2  |-  ( ph  ->  ( ( cos `  A
)  -  ( cos `  B ) )  e.  RR+ )
786recoscld 11437 . . 3  |-  ( ph  ->  ( cos `  B
)  e.  RR )
7911recoscld 11437 . . 3  |-  ( ph  ->  ( cos `  A
)  e.  RR )
80 difrp 9487 . . 3  |-  ( ( ( cos `  B
)  e.  RR  /\  ( cos `  A )  e.  RR )  -> 
( ( cos `  B
)  <  ( cos `  A )  <->  ( ( cos `  A )  -  ( cos `  B ) )  e.  RR+ )
)
8178, 79, 80syl2anc 408 . 2  |-  ( ph  ->  ( ( cos `  B
)  <  ( cos `  A )  <->  ( ( cos `  A )  -  ( cos `  B ) )  e.  RR+ )
)
8277, 81mpbird 166 1  |-  ( ph  ->  ( cos `  B
)  <  ( cos `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   RRcr 7626   0cc0 7627    + caddc 7630    x. cmul 7632   RR*cxr 7806    < clt 7807    <_ cle 7808    - cmin 7940    / cdiv 8439   2c2 8778   RR+crp 9448   (,)cioo 9678   [,]cicc 9681   sincsin 11357   cosccos 11358   picpi 11360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747  ax-pre-suploc 7748  ax-addf 7749  ax-mulf 7750
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-9 8793  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-ioo 9682  df-ioc 9683  df-ico 9684  df-icc 9685  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-bc 10501  df-ihash 10529  df-shft 10594  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364  df-pi 11366  df-rest 12131  df-topgen 12150  df-psmet 12165  df-xmet 12166  df-met 12167  df-bl 12168  df-mopn 12169  df-top 12174  df-topon 12187  df-bases 12219  df-ntr 12274  df-cn 12366  df-cnp 12367  df-tx 12431  df-cncf 12736  df-limced 12803  df-dvap 12804
This theorem is referenced by:  cosq34lt1  12947  cos02pilt1  12948  cos0pilt1  12949  cos11  12950  ioocosf1o  12951
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