Theorem cosordlem 13410

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 7899	. . . . . . . 8 |- 0 e. RR
3		pire 13347	. . . . . . . 8 |- pi e. RR
4	2, 3	elicc2i 9875	. . . . . . 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 999	. . . . 5 |- ( ph -> B e. RR )
7	6	recnd 7927	. . . 4 |- ( ph -> B e. CC )
8		cosord.1	. . . . . . 7 |- ( ph -> A e. ( 0 [,] pi ) )
9	2, 3	elicc2i 9875	. . . . . . 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 999	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 7927	. . . 4 |- ( ph -> A e. CC )
13		subcos 11688	. . . 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 9594	. . . 4 |- 2 e. RR+
16	6, 11	readdcld 7928	. . . . . . . 8 |- ( ph -> ( B + A ) e. RR )
17	16	rehalfcld 9103	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. RR )
18	17	resincld 11664	. . . . . 6 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR )
19	2	a1i 9	. . . . . . . . . . 11 |- ( ph -> 0 e. RR )
20	10	simp2d 1000	. . . . . . . . . . 11 |- ( ph -> 0 <_ A )
21		cosord.3	. . . . . . . . . . 11 |- ( ph -> A < B )
22	19, 11, 6, 20, 21	lelttrd 8023	. . . . . . . . . 10 |- ( ph -> 0 < B )
23	6, 11, 22, 20	addgtge0d 8418	. . . . . . . . 9 |- ( ph -> 0 < ( B + A ) )
24		2re 8927	. . . . . . . . . 10 |- 2 e. RR
25		2pos 8948	. . . . . . . . . 10 |- 0 < 2
26		divgt0 8767	. . . . . . . . . 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 8320	. . . . . . . . . . 11 |- ( ph -> ( B + A ) < ( B + B ) )
31	7	2timesd 9099	. . . . . . . . . . 11 |- ( ph -> ( 2 x. B ) = ( B + B ) )
32	30, 31	breqtrrd 4010	. . . . . . . . . 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 8771	. . . . . . . . . . 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 1232	. . . . . . . . . 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 1001	. . . . . . . . 9 |- ( ph -> B <_ pi )
39	17, 6, 29, 37, 38	ltletrd 8321	. . . . . . . 8 |- ( ph -> ( ( B + A ) / 2 ) < pi )
40		0xr 7945	. . . . . . . . 9 |- 0 e. RR*
41	3	rexri 7956	. . . . . . . . 9 |- pi e. RR*
42		elioo2 9857	. . . . . . . . 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 1171	. . . . . . 7 |- ( ph -> ( ( B + A ) / 2 ) e. ( 0 (,) pi ) )
45		sinq12gt0 13391	. . . . . . 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 9629	. . . . 5 |- ( ph -> ( sin ` ( ( B + A ) / 2 ) ) e. RR+ )
48	6, 11	resubcld 8279	. . . . . . . 8 |- ( ph -> ( B - A ) e. RR )
49	48	rehalfcld 9103	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. RR )
50	49	resincld 11664	. . . . . 6 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR )
51	11, 6	posdifd 8430	. . . . . . . . . 10 |- ( ph -> ( A < B <-> 0 < ( B - A ) ) )
52	21, 51	mpbid 146	. . . . . . . . 9 |- ( ph -> 0 < ( B - A ) )
53		divgt0 8767	. . . . . . . . . 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 9084	. . . . . . . . . 10 |- ( pi e. RR -> ( pi / 2 ) e. RR )
57	3, 56	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) e. RR )
58	6, 11	subge02d 8435	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ A <-> ( B - A ) <_ B ) )
59	20, 58	mpbid 146	. . . . . . . . . . 11 |- ( ph -> ( B - A ) <_ B )
60	48, 6, 29, 59, 38	letrd 8022	. . . . . . . . . 10 |- ( ph -> ( B - A ) <_ pi )
61		lediv1 8764	. . . . . . . . . . 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 1232	. . . . . . . . . 10 |- ( ph -> ( ( B - A ) <_ pi <-> ( ( B - A ) / 2 ) <_ ( pi / 2 ) ) )
63	60, 62	mpbid 146	. . . . . . . . 9 |- ( ph -> ( ( B - A ) / 2 ) <_ ( pi / 2 ) )
64		pirp 13350	. . . . . . . . . 10 |- pi e. RR+
65		rphalflt 9619	. . . . . . . . . 10 |- ( pi e. RR+ -> ( pi / 2 ) < pi )
66	64, 65	mp1i 10	. . . . . . . . 9 |- ( ph -> ( pi / 2 ) < pi )
67	49, 57, 29, 63, 66	lelttrd 8023	. . . . . . . 8 |- ( ph -> ( ( B - A ) / 2 ) < pi )
68		elioo2 9857	. . . . . . . . 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 1171	. . . . . . 7 |- ( ph -> ( ( B - A ) / 2 ) e. ( 0 (,) pi ) )
71		sinq12gt0 13391	. . . . . . 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 9629	. . . . 5 |- ( ph -> ( sin ` ( ( B - A ) / 2 ) ) e. RR+ )
74	47, 73	rpmulcld 9649	. . . 4 |- ( ph -> ( ( sin ` ( ( B + A ) / 2 ) ) x. ( sin ` ( ( B - A ) / 2 ) ) ) e. RR+ )
75		rpmulcl 9614	. . . 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 2243	. 2 |- ( ph -> ( ( cos ` A ) - ( cos ` B ) ) e. RR+ )
78	6	recoscld 11665	. . 3 |- ( ph -> ( cos ` B ) e. RR )
79	11	recoscld 11665	. . 3 |- ( ph -> ( cos ` A ) e. RR )
80		difrp 9628	. . 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 968   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   + caddc 7756   x. cmul 7758   RR*cxr 7932   < clt 7933   <_ cle 7934   - cmin 8069   / cdiv 8568   2c2 8908   RR+crp 9589   (,)cioo 9824   [,]cicc 9827   sincsin 11585   cosccos 11586   picpi 11588

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-5 8919  df-6 8920  df-7 8921  df-8 8922  df-9 8923  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ioc 9829  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591  df-cos 11592  df-pi 11594  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266

This theorem is referenced by:  cosq34lt1  13411  cos02pilt1  13412  cos0pilt1  13413  cos11  13414  ioocosf1o  13415