Theorem cosordlem 12978

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 7790	. . . . . . . 8 |- 0 e. RR
3		pire 12915	. . . . . . . 8 |- pi e. RR
4	2, 3	elicc2i 9752	. . . . . . 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 994	. . . . 5 |- ( ph -> B e. RR )
7	6	recnd 7818	. . . 4 |- ( ph -> B e. CC )
8		cosord.1	. . . . . . 7 |- ( ph -> A e. ( 0 [,] pi ) )
9	2, 3	elicc2i 9752	. . . . . . 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 994	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 7818	. . . 4 |- ( ph -> A e. CC )
13		subcos 11490	. . . 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 409	. . 3 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
15		2rp 9475	. . . 4 |- 2 e. RR+
16	6, 11	readdcld 7819	. . . . . . . 8 |- ( ph -> ( B + A ) e. RR )
17	16	rehalfcld 8990	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. RR )
18	17	resincld 11466	. . . . . 6 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR )
19	2	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 e. RR )
20	10	simp2d 995	. . . . . . . . . . 11 |- ( ph -> 0 <_ A )
21		cosord.3	. . . . . . . . . . 11 |- ( ph -> A < B )
22	19, 11, 6, 20, 21	lelttrd 7911	. . . . . . . . . 10 |- ( ph -> 0 < B )
23	6, 11, 22, 20	addgtge0d 8306	. . . . . . . . 9 |- ( ph -> 0 < ( B + A ) )
24		2re 8814	. . . . . . . . . 10 |- 2 e. RR
25		2pos 8835	. . . . . . . . . 10 |- 0 < 2
26		divgt0 8654	. . . . . . . . . 10 |- ( ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B + A ) / 2 ) )
27	24, 25, 26	mpanr12 436	. . . . . . . . 9 |- ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) -> 0 < ( ( B + A ) / 2 ) )
28	16, 23, 27	syl2anc 409	. . . . . . . 8 |- ( ph -> 0 < ( ( B + A ) / 2 ) )
29	3	a1i 9	. . . . . . . . 9 |- ( ph -> pi e. RR )
30	11, 6, 6, 21	ltadd2dd 8208	. . . . . . . . . . 11 |- ( ph -> ( B + A ) < ( B + B ) )
31	7	2timesd 8986	. . . . . . . . . . 11 |- ( ph -> ( 2 x. B ) = ( B + B ) )
32	30, 31	breqtrrd 3964	. . . . . . . . . 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 8658	. . . . . . . . . . 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 1221	. . . . . . . . . 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 996	. . . . . . . . 9 |- ( ph -> B <_ pi )
39	17, 6, 29, 37, 38	ltletrd 8209	. . . . . . . 8 |- ( ph -> ( ( B + A ) / 2 ) < pi )
40		0xr 7836	. . . . . . . . 9 |- 0 e. RR*
41	3	rexri 7847	. . . . . . . . 9 |- pi e. RR*
42		elioo2 9734	. . . . . . . . 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 423	. . . . . . . 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 1166	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) pi ) )
45		sinq12gt0 12959	. . . . . . 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 9510	. . . . 5 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ )
48	6, 11	resubcld 8167	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	48	rehalfcld 8990	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. RR )
50	49	resincld 11466	. . . . . 6 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
51	11, 6	posdifd 8318	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
52	21, 51	mpbid 146	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
53		divgt0 8654	. . . . . . . . . 10 |- ( ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B - A ) / 2 ) )
54	24, 25, 53	mpanr12 436	. . . . . . . . 9 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> 0 < ( ( B - A ) / 2 ) )
55	48, 52, 54	syl2anc 409	. . . . . . . 8 |- ( ph -> 0 < ( ( B - A ) / 2 ) )
56		rehalfcl 8971	. . . . . . . . . 10 |- ( pi e. RR -> ( pi / 2 ) e. RR )
57	3, 56	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) e. RR )
58	6, 11	subge02d 8323	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) )
59	20, 58	mpbid 146	. . . . . . . . . . 11 |- ( ph -> ( B - A ) <_ B )
60	48, 6, 29, 59, 38	letrd 7910	. . . . . . . . . 10 |- ( ph -> ( B - A ) <_ pi )
61		lediv1 8651	. . . . . . . . . . 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 1221	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
63	60, 62	mpbid 146	. . . . . . . . 9 |- ( ph -> ( ( B - A ) / 2 ) <_ ( pi / 2 ) )
64		pirp 12918	. . . . . . . . . 10 |- pi e. RR+
65		rphalflt 9500	. . . . . . . . . 10 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
66	64, 65	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) < pi )
67	49, 57, 29, 63, 66	lelttrd 7911	. . . . . . . 8 |- ( ph -> ( ( B - A ) / 2 ) < pi )
68		elioo2 9734	. . . . . . . . 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 423	. . . . . . . 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 1166	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) pi ) )
71		sinq12gt0 12959	. . . . . . 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 9510	. . . . 5 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ )
74	47, 73	rpmulcld 9530	. . . 4 |- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ )
75		rpmulcl 9495	. . . 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 411	. . 3 |- ( ph -> ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) e. RR+ )
77	14, 76	eqeltrd 2217	. 2 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ )
78	6	recoscld 11467	. . 3 |- ( ph -> ( cos ` B ) e. RR )
79	11	recoscld 11467	. . 3 |- ( ph -> ( cos ` A ) e. RR )
80		difrp 9509	. . 3 |- ( ( ( cos ` B ) e. RR /\ ( cos ` A ) e. RR ) -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) )
81	78, 79, 80	syl2anc 409	. 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 963   = wceq 1332   e. wcel 1481   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   + caddc 7647   x. cmul 7649   RR*cxr 7823   < clt 7824   <_ cle 7825   - cmin 7957   / cdiv 8456   2c2 8795   RR+crp 9470   (,)cioo 9701   [,]cicc 9704   sincsin 11387   cosccos 11388   picpi 11390

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764  ax-pre-suploc 7765  ax-addf 7766  ax-mulf 7767

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-disj 3915  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-map 6552  df-pm 6553  df-en 6643  df-dom 6644  df-fin 6645  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-7 8808  df-8 8809  df-9 8810  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-xneg 9589  df-xadd 9590  df-ioo 9705  df-ioc 9706  df-ico 9707  df-icc 9708  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-shft 10619  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391  df-sin 11393  df-cos 11394  df-pi 11396  df-rest 12161  df-topgen 12180  df-psmet 12195  df-xmet 12196  df-met 12197  df-bl 12198  df-mopn 12199  df-top 12204  df-topon 12217  df-bases 12249  df-ntr 12304  df-cn 12396  df-cnp 12397  df-tx 12461  df-cncf 12766  df-limced 12833  df-dvap 12834

This theorem is referenced by:  cosq34lt1  12979  cos02pilt1  12980  cos0pilt1  12981  cos11  12982  ioocosf1o  12983