Theorem cosordlem 12930

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

Step	Hyp	Ref	Expression
1		cosord.2	. . . . . . 7 |- ( ph -> B e. ( 0 [,] pi ) )
2		0re 7766	. . . . . . . 8 |- 0 e. RR
3		pire 12867	. . . . . . . 8 |- pi e. RR
4	2, 3	elicc2i 9722	. . . . . . 7 |- ( B e. ( 0 [,] pi ) <-> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
5	1, 4	sylib 121	. . . . . 6 |- ( ph -> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
6	5	simp1d 993	. . . . 5 |- ( ph -> B e. RR )
7	6	recnd 7794	. . . 4 |- ( ph -> B e. CC )
8		cosord.1	. . . . . . 7 |- ( ph -> A e. ( 0 [,] pi ) )
9	2, 3	elicc2i 9722	. . . . . . 7 |- ( A e. ( 0 [,] pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
10	8, 9	sylib 121	. . . . . 6 |- ( ph -> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
11	10	simp1d 993	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 7794	. . . 4 |- ( ph -> A e. CC )
13		subcos 11454	. . . 4 |- ( ( B e. CC /\ A e. CC ) -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
14	7, 12, 13	syl2anc 408	. . 3 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
15		2rp 9446	. . . 4 |- 2 e. RR+
16	6, 11	readdcld 7795	. . . . . . . 8 |- ( ph -> ( B + A ) e. RR )
17	16	rehalfcld 8966	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. RR )
18	17	resincld 11430	. . . . . 6 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR )
19	2	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 e. RR )
20	10	simp2d 994	. . . . . . . . . . 11 |- ( ph -> 0 <_ A )
21		cosord.3	. . . . . . . . . . 11 |- ( ph -> A < B )
22	19, 11, 6, 20, 21	lelttrd 7887	. . . . . . . . . 10 |- ( ph -> 0 < B )
23	6, 11, 22, 20	addgtge0d 8282	. . . . . . . . 9 |- ( ph -> 0 < ( B + A ) )
24		2re 8790	. . . . . . . . . 10 |- 2 e. RR
25		2pos 8811	. . . . . . . . . 10 |- 0 < 2
26		divgt0 8630	. . . . . . . . . 10 |- ( ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B + A ) / 2 ) )
27	24, 25, 26	mpanr12 435	. . . . . . . . 9 |- ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) -> 0 < ( ( B + A ) / 2 ) )
28	16, 23, 27	syl2anc 408	. . . . . . . 8 |- ( ph -> 0 < ( ( B + A ) / 2 ) )
29	3	a1i 9	. . . . . . . . 9 |- ( ph -> pi e. RR )
30	11, 6, 6, 21	ltadd2dd 8184	. . . . . . . . . . 11 |- ( ph -> ( B + A ) < ( B + B ) )
31	7	2timesd 8962	. . . . . . . . . . 11 |- ( ph -> ( 2 x. B ) = ( B + B ) )
32	30, 31	breqtrrd 3956	. . . . . . . . . 10 |- ( ph -> ( B + A ) < ( 2 x. B ) )
33	24	a1i 9	. . . . . . . . . . 11 |- ( ph -> 2 e. RR )
34	25	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 < 2 )
35		ltdivmul 8634	. . . . . . . . . . 11 |- ( ( ( B + A ) e. RR /\ B e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( ( B + A ) / 2 ) < B <-> ( B + A ) < ( 2 x. B ) ) )
36	16, 6, 33, 34, 35	syl112anc 1220	. . . . . . . . . 10 |- ( ph -> ( ( ( B + A ) / 2 ) < B <-> ( B + A ) < ( 2 x. B ) ) )
37	32, 36	mpbird 166	. . . . . . . . 9 |- ( ph -> ( ( B + A ) / 2 ) < B )
38	5	simp3d 995	. . . . . . . . 9 |- ( ph -> B <_ pi )
39	17, 6, 29, 37, 38	ltletrd 8185	. . . . . . . 8 |- ( ph -> ( ( B + A ) / 2 ) < pi )
40		0xr 7812	. . . . . . . . 9 |- 0 e. RR*
41	3	rexri 7823	. . . . . . . . 9 |- pi e. RR*
42		elioo2 9704	. . . . . . . . 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 ) ) )
43	40, 41, 42	mp2an 422	. . . . . . . 8 |- ( ( ( B + A ) / 2 ) e. ( 0 (,) pi ) <-> ( ( ( B + A ) / 2 ) e. RR /\ 0 < ( ( B + A ) / 2 ) /\ ( ( B + A ) / 2 ) < pi ) )
44	17, 28, 39, 43	syl3anbrc 1165	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) pi ) )
45		sinq12gt0 12911	. . . . . . 7 |- ( ( ( B + A ) / 2 ) e. ( 0 (,) pi ) -> 0 < ( sin ` ( ( B + A ) / 2 ) ) )
46	44, 45	syl 14	. . . . . 6 |- ( ph -> 0 < ( sin ` ( ( B + A ) / 2 ) ) )
47	18, 46	elrpd 9481	. . . . 5 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ )
48	6, 11	resubcld 8143	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	48	rehalfcld 8966	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. RR )
50	49	resincld 11430	. . . . . 6 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
51	11, 6	posdifd 8294	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
52	21, 51	mpbid 146	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
53		divgt0 8630	. . . . . . . . . 10 |- ( ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B - A ) / 2 ) )
54	24, 25, 53	mpanr12 435	. . . . . . . . 9 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> 0 < ( ( B - A ) / 2 ) )
55	48, 52, 54	syl2anc 408	. . . . . . . 8 |- ( ph -> 0 < ( ( B - A ) / 2 ) )
56		rehalfcl 8947	. . . . . . . . . 10 |- ( pi e. RR -> ( pi / 2 ) e. RR )
57	3, 56	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) e. RR )
58	6, 11	subge02d 8299	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) )
59	20, 58	mpbid 146	. . . . . . . . . . 11 |- ( ph -> ( B - A ) <_ B )
60	48, 6, 29, 59, 38	letrd 7886	. . . . . . . . . 10 |- ( ph -> ( B - A ) <_ pi )
61		lediv1 8627	. . . . . . . . . . 11 |- ( ( ( B - A ) e. RR /\ pi e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
62	48, 29, 33, 34, 61	syl112anc 1220	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
63	60, 62	mpbid 146	. . . . . . . . 9 |- ( ph -> ( ( B - A ) / 2 ) <_ ( pi / 2 ) )
64		pirp 12870	. . . . . . . . . 10 |- pi e. RR+
65		rphalflt 9471	. . . . . . . . . 10 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
66	64, 65	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) < pi )
67	49, 57, 29, 63, 66	lelttrd 7887	. . . . . . . 8 |- ( ph -> ( ( B - A ) / 2 ) < pi )
68		elioo2 9704	. . . . . . . . 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 ) ) )
69	40, 41, 68	mp2an 422	. . . . . . . 8 |- ( ( ( B - A ) / 2 ) e. ( 0 (,) pi ) <-> ( ( ( B - A ) / 2 ) e. RR /\ 0 < ( ( B - A ) / 2 ) /\ ( ( B - A ) / 2 ) < pi ) )
70	49, 55, 67, 69	syl3anbrc 1165	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) pi ) )
71		sinq12gt0 12911	. . . . . . 7 |- ( ( ( B - A ) / 2 ) e. ( 0 (,) pi ) -> 0 < ( sin ` ( ( B - A ) / 2 ) ) )
72	70, 71	syl 14	. . . . . 6 |- ( ph -> 0 < ( sin ` ( ( B - A ) / 2 ) ) )
73	50, 72	elrpd 9481	. . . . 5 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ )
74	47, 73	rpmulcld 9500	. . . 4 |- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ )
75		rpmulcl 9466	. . . 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+ )
76	15, 74, 75	sylancr 410	. . 3 |- ( ph -> ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) e. RR+ )
77	14, 76	eqeltrd 2216	. 2 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ )
78	6	recoscld 11431	. . 3 |- ( ph -> ( cos ` B ) e. RR )
79	11	recoscld 11431	. . 3 |- ( ph -> ( cos ` A ) e. RR )
80		difrp 9480	. . 3 |- ( ( ( cos ` B ) e. RR /\ ( cos ` A ) e. RR ) -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) )
81	78, 79, 80	syl2anc 408	. 2 |- ( ph -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) )
82	77, 81	mpbird 166	1 |- ( ph -> ( cos ` B ) < ( cos ` A ) )

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 7618   RRcr 7619   0cc0 7620   + caddc 7623   x. cmul 7625   RR*cxr 7799   < clt 7800   <_ cle 7801   - cmin 7933   / cdiv 8432   2c2 8771   RR+crp 9441   (,)cioo 9671   [,]cicc 9674   sincsin 11350   cosccos 11351   picpi 11353

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 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740  ax-pre-suploc 7741  ax-addf 7742  ax-mulf 7743

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 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-5 8782  df-6 8783  df-7 8784  df-8 8785  df-9 8786  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-ioo 9675  df-ioc 9676  df-ico 9677  df-icc 9678  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-fac 10472  df-bc 10494  df-ihash 10522  df-shft 10587  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123  df-ef 11354  df-sin 11356  df-cos 11357  df-pi 11359  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-tx 12422  df-cncf 12727  df-limced 12794  df-dvap 12795

This theorem is referenced by:  cosq34lt1  12931  cos02pilt1  12932