Theorem cosordlem 15575

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 8179	. . . . . . . 8 |- 0 e. RR
3		pire 15512	. . . . . . . 8 |- pi e. RR
4	2, 3	elicc2i 10174	. . . . . . 7 |- ( B e. ( 0 [,] pi ) <-> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
5	1, 4	sylib 122	. . . . . 6 |- ( ph -> ( B e. RR /\ 0 <_ B /\ B <_ pi ) )
6	5	simp1d 1035	. . . . 5 |- ( ph -> B e. RR )
7	6	recnd 8208	. . . 4 |- ( ph -> B e. CC )
8		cosord.1	. . . . . . 7 |- ( ph -> A e. ( 0 [,] pi ) )
9	2, 3	elicc2i 10174	. . . . . . 7 |- ( A e. ( 0 [,] pi ) <-> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
10	8, 9	sylib 122	. . . . . 6 |- ( ph -> ( A e. RR /\ 0 <_ A /\ A <_ pi ) )
11	10	simp1d 1035	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 8208	. . . 4 |- ( ph -> A e. CC )
13		subcos 12309	. . . 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 411	. . 3 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) = ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) )
15		2rp 9893	. . . 4 |- 2 e. RR+
16	6, 11	readdcld 8209	. . . . . . . 8 |- ( ph -> ( B + A ) e. RR )
17	16	rehalfcld 9391	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. RR )
18	17	resincld 12285	. . . . . 6 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR )
19	2	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 e. RR )
20	10	simp2d 1036	. . . . . . . . . . 11 |- ( ph -> 0 <_ A )
21		cosord.3	. . . . . . . . . . 11 |- ( ph -> A < B )
22	19, 11, 6, 20, 21	lelttrd 8304	. . . . . . . . . 10 |- ( ph -> 0 < B )
23	6, 11, 22, 20	addgtge0d 8700	. . . . . . . . 9 |- ( ph -> 0 < ( B + A ) )
24		2re 9213	. . . . . . . . . 10 |- 2 e. RR
25		2pos 9234	. . . . . . . . . 10 |- 0 < 2
26		divgt0 9052	. . . . . . . . . 10 |- ( ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B + A ) / 2 ) )
27	24, 25, 26	mpanr12 439	. . . . . . . . 9 |- ( ( ( B + A ) e. RR /\ 0 < ( B + A ) ) -> 0 < ( ( B + A ) / 2 ) )
28	16, 23, 27	syl2anc 411	. . . . . . . 8 |- ( ph -> 0 < ( ( B + A ) / 2 ) )
29	3	a1i 9	. . . . . . . . 9 |- ( ph -> pi e. RR )
30	11, 6, 6, 21	ltadd2dd 8602	. . . . . . . . . . 11 |- ( ph -> ( B + A ) < ( B + B ) )
31	7	2timesd 9387	. . . . . . . . . . 11 |- ( ph -> ( 2 x. B ) = ( B + B ) )
32	30, 31	breqtrrd 4116	. . . . . . . . . 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 9056	. . . . . . . . . . 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 1277	. . . . . . . . . 10 |- ( ph -> ( ( ( B + A ) / 2 ) < B <-> ( B + A ) < ( 2 x. B ) ) )
37	32, 36	mpbird 167	. . . . . . . . 9 |- ( ph -> ( ( B + A ) / 2 ) < B )
38	5	simp3d 1037	. . . . . . . . 9 |- ( ph -> B <_ pi )
39	17, 6, 29, 37, 38	ltletrd 8603	. . . . . . . 8 |- ( ph -> ( ( B + A ) / 2 ) < pi )
40		0xr 8226	. . . . . . . . 9 |- 0 e. RR*
41	3	rexri 8237	. . . . . . . . 9 |- pi e. RR*
42		elioo2 10156	. . . . . . . . 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 426	. . . . . . . 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 1207	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) pi ) )
45		sinq12gt0 15556	. . . . . . 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 9928	. . . . 5 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ )
48	6, 11	resubcld 8560	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	48	rehalfcld 9391	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. RR )
50	49	resincld 12285	. . . . . 6 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
51	11, 6	posdifd 8712	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
52	21, 51	mpbid 147	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
53		divgt0 9052	. . . . . . . . . 10 |- ( ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) /\ ( 2 e. RR /\ 0 < 2 ) ) -> 0 < ( ( B - A ) / 2 ) )
54	24, 25, 53	mpanr12 439	. . . . . . . . 9 |- ( ( ( B - A ) e. RR /\ 0 < ( B - A ) ) -> 0 < ( ( B - A ) / 2 ) )
55	48, 52, 54	syl2anc 411	. . . . . . . 8 |- ( ph -> 0 < ( ( B - A ) / 2 ) )
56		rehalfcl 9371	. . . . . . . . . 10 |- ( pi e. RR -> ( pi / 2 ) e. RR )
57	3, 56	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) e. RR )
58	6, 11	subge02d 8717	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) )
59	20, 58	mpbid 147	. . . . . . . . . . 11 |- ( ph -> ( B - A ) <_ B )
60	48, 6, 29, 59, 38	letrd 8303	. . . . . . . . . 10 |- ( ph -> ( B - A ) <_ pi )
61		lediv1 9049	. . . . . . . . . . 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 1277	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
63	60, 62	mpbid 147	. . . . . . . . 9 |- ( ph -> ( ( B - A ) / 2 ) <_ ( pi / 2 ) )
64		pirp 15515	. . . . . . . . . 10 |- pi e. RR+
65		rphalflt 9918	. . . . . . . . . 10 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
66	64, 65	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) < pi )
67	49, 57, 29, 63, 66	lelttrd 8304	. . . . . . . 8 |- ( ph -> ( ( B - A ) / 2 ) < pi )
68		elioo2 10156	. . . . . . . . 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 426	. . . . . . . 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 1207	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) pi ) )
71		sinq12gt0 15556	. . . . . . 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 9928	. . . . 5 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ )
74	47, 73	rpmulcld 9948	. . . 4 |- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ )
75		rpmulcl 9913	. . . 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 414	. . 3 |- ( ph -> ( 2 x. ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) ) e. RR+ )
77	14, 76	eqeltrd 2308	. 2 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ )
78	6	recoscld 12286	. . 3 |- ( ph -> ( cos ` B ) e. RR )
79	11	recoscld 12286	. . 3 |- ( ph -> ( cos ` A ) e. RR )
80		difrp 9927	. . 3 |- ( ( ( cos ` B ) e. RR /\ ( cos ` A ) e. RR ) -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) )
81	78, 79, 80	syl2anc 411	. 2 |- ( ph -> ( ( cos ` B ) < ( cos ` A ) <-> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ ) )
82	77, 81	mpbird 167	1 |- ( ph -> ( cos ` B ) < ( cos ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   + caddc 8035   x. cmul 8037   RR*cxr 8213   < clt 8214   <_ cle 8215   - cmin 8350   / cdiv 8852   2c2 9194   RR+crp 9888   (,)cioo 10123   [,]cicc 10126   sincsin 12206   cosccos 12207   picpi 12209

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-pre-suploc 8153  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-disj 4065  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-map 6819  df-pm 6820  df-en 6910  df-dom 6911  df-fin 6912  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-ioo 10127  df-ioc 10128  df-ico 10129  df-icc 10130  df-fz 10244  df-fzo 10378  df-seqfrec 10710  df-exp 10801  df-fac 10988  df-bc 11010  df-ihash 11038  df-shft 11376  df-cj 11403  df-re 11404  df-im 11405  df-rsqrt 11559  df-abs 11560  df-clim 11840  df-sumdc 11915  df-ef 12210  df-sin 12212  df-cos 12213  df-pi 12215  df-rest 13325  df-topgen 13344  df-psmet 14559  df-xmet 14560  df-met 14561  df-bl 14562  df-mopn 14563  df-top 14724  df-topon 14737  df-bases 14769  df-ntr 14822  df-cn 14914  df-cnp 14915  df-tx 14979  df-cncf 15297  df-limced 15382  df-dvap 15383

This theorem is referenced by:  cosq34lt1  15576  cos02pilt1  15577  cos0pilt1  15578  cos11  15579  ioocosf1o  15580