Theorem cosordlem 15321

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 8072	. . . . . . . 8 |- 0 e. RR
3		pire 15258	. . . . . . . 8 |- pi e. RR
4	2, 3	elicc2i 10061	. . . . . . 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 1012	. . . . 5 |- ( ph -> B e. RR )
7	6	recnd 8101	. . . 4 |- ( ph -> B e. CC )
8		cosord.1	. . . . . . 7 |- ( ph -> A e. ( 0 [,] pi ) )
9	2, 3	elicc2i 10061	. . . . . . 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 1012	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 8101	. . . 4 |- ( ph -> A e. CC )
13		subcos 12058	. . . 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 9780	. . . 4 |- 2 e. RR+
16	6, 11	readdcld 8102	. . . . . . . 8 |- ( ph -> ( B + A ) e. RR )
17	16	rehalfcld 9284	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. RR )
18	17	resincld 12034	. . . . . 6 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR )
19	2	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 e. RR )
20	10	simp2d 1013	. . . . . . . . . . 11 |- ( ph -> 0 <_ A )
21		cosord.3	. . . . . . . . . . 11 |- ( ph -> A < B )
22	19, 11, 6, 20, 21	lelttrd 8197	. . . . . . . . . 10 |- ( ph -> 0 < B )
23	6, 11, 22, 20	addgtge0d 8593	. . . . . . . . 9 |- ( ph -> 0 < ( B + A ) )
24		2re 9106	. . . . . . . . . 10 |- 2 e. RR
25		2pos 9127	. . . . . . . . . 10 |- 0 < 2
26		divgt0 8945	. . . . . . . . . 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 8495	. . . . . . . . . . 11 |- ( ph -> ( B + A ) < ( B + B ) )
31	7	2timesd 9280	. . . . . . . . . . 11 |- ( ph -> ( 2 x. B ) = ( B + B ) )
32	30, 31	breqtrrd 4072	. . . . . . . . . 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 8949	. . . . . . . . . . 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 1254	. . . . . . . . . 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 1014	. . . . . . . . 9 |- ( ph -> B <_ pi )
39	17, 6, 29, 37, 38	ltletrd 8496	. . . . . . . 8 |- ( ph -> ( ( B + A ) / 2 ) < pi )
40		0xr 8119	. . . . . . . . 9 |- 0 e. RR*
41	3	rexri 8130	. . . . . . . . 9 |- pi e. RR*
42		elioo2 10043	. . . . . . . . 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 1184	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) pi ) )
45		sinq12gt0 15302	. . . . . . 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 9815	. . . . 5 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ )
48	6, 11	resubcld 8453	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	48	rehalfcld 9284	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. RR )
50	49	resincld 12034	. . . . . 6 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
51	11, 6	posdifd 8605	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
52	21, 51	mpbid 147	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
53		divgt0 8945	. . . . . . . . . 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 9264	. . . . . . . . . 10 |- ( pi e. RR -> ( pi / 2 ) e. RR )
57	3, 56	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) e. RR )
58	6, 11	subge02d 8610	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) )
59	20, 58	mpbid 147	. . . . . . . . . . 11 |- ( ph -> ( B - A ) <_ B )
60	48, 6, 29, 59, 38	letrd 8196	. . . . . . . . . 10 |- ( ph -> ( B - A ) <_ pi )
61		lediv1 8942	. . . . . . . . . . 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 1254	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
63	60, 62	mpbid 147	. . . . . . . . 9 |- ( ph -> ( ( B - A ) / 2 ) <_ ( pi / 2 ) )
64		pirp 15261	. . . . . . . . . 10 |- pi e. RR+
65		rphalflt 9805	. . . . . . . . . 10 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
66	64, 65	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) < pi )
67	49, 57, 29, 63, 66	lelttrd 8197	. . . . . . . 8 |- ( ph -> ( ( B - A ) / 2 ) < pi )
68		elioo2 10043	. . . . . . . . 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 1184	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) pi ) )
71		sinq12gt0 15302	. . . . . . 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 9815	. . . . 5 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ )
74	47, 73	rpmulcld 9835	. . . 4 |- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ )
75		rpmulcl 9800	. . . 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 2282	. 2 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ )
78	6	recoscld 12035	. . 3 |- ( ph -> ( cos ` B ) e. RR )
79	11	recoscld 12035	. . 3 |- ( ph -> ( cos ` A ) e. RR )
80		difrp 9814	. . 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 981   = wceq 1373   e. wcel 2176   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   + caddc 7928   x. cmul 7930   RR*cxr 8106   < clt 8107   <_ cle 8108   - cmin 8243   / cdiv 8745   2c2 9087   RR+crp 9775   (,)cioo 10010   [,]cicc 10013   sincsin 11955   cosccos 11956   picpi 11958

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045  ax-pre-suploc 8046  ax-addf 8047  ax-mulf 8048

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-of 6158  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-map 6737  df-pm 6738  df-en 6828  df-dom 6829  df-fin 6830  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-9 9102  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-xneg 9894  df-xadd 9895  df-ioo 10014  df-ioc 10015  df-ico 10016  df-icc 10017  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-fac 10871  df-bc 10893  df-ihash 10921  df-shft 11126  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665  df-ef 11959  df-sin 11961  df-cos 11962  df-pi 11964  df-rest 13073  df-topgen 13092  df-psmet 14305  df-xmet 14306  df-met 14307  df-bl 14308  df-mopn 14309  df-top 14470  df-topon 14483  df-bases 14515  df-ntr 14568  df-cn 14660  df-cnp 14661  df-tx 14725  df-cncf 15043  df-limced 15128  df-dvap 15129

This theorem is referenced by:  cosq34lt1  15322  cos02pilt1  15323  cos0pilt1  15324  cos11  15325  ioocosf1o  15326