Theorem cos01gt0 11505

Description: The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Assertion

Ref	Expression
cos01gt0	|- ( A e. ( 0 (,] 1 ) -> 0 < ( cos ` A ) )

Proof of Theorem cos01gt0

Step	Hyp	Ref	Expression
1		0xr 7836	. . . . . . . . . 10 |- 0 e. RR*
2		1re 7789	. . . . . . . . . 10 |- 1 e. RR
3		elioc2 9749	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 423	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 997	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6	5	resqcld 10481	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. RR )
7	6	recnd 7818	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. CC )
8		2cn 8815	. . . . . . 7 |- 2 e. CC
9		3cn 8819	. . . . . . . 8 |- 3 e. CC
10		3ap0 8840	. . . . . . . 8 |- 3 # 0
11	9, 10	pm3.2i 270	. . . . . . 7 |- ( 3 e. CC /\ 3 # 0 )
12		div12ap 8478	. . . . . . 7 |- ( ( 2 e. CC /\ ( A ^ 2 ) e. CC /\ ( 3 e. CC /\ 3 # 0 ) ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
13	8, 11, 12	mp3an13 1307	. . . . . 6 |- ( ( A ^ 2 ) e. CC -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
14	7, 13	syl 14	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
15		2z 9106	. . . . . . . . . 10 |- 2 e. ZZ
16		expgt0 10357	. . . . . . . . . 10 |- ( ( A e. RR /\ 2 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 2 ) )
17	15, 16	mp3an2 1304	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 2 ) )
18	17	3adant3 1002	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> 0 < ( A ^ 2 ) )
19	4, 18	sylbi 120	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A ^ 2 ) )
20		2lt3 8914	. . . . . . . . . 10 |- 2 < 3
21		2re 8814	. . . . . . . . . . 11 |- 2 e. RR
22		3re 8818	. . . . . . . . . . 11 |- 3 e. RR
23		3pos 8838	. . . . . . . . . . 11 |- 0 < 3
24	21, 22, 22, 23	ltdiv1ii 8711	. . . . . . . . . 10 |- ( 2 < 3 <-> ( 2 / 3 ) < ( 3 / 3 ) )
25	20, 24	mpbi 144	. . . . . . . . 9 |- ( 2 / 3 ) < ( 3 / 3 )
26	9, 10	dividapi 8529	. . . . . . . . 9 |- ( 3 / 3 ) = 1
27	25, 26	breqtri 3961	. . . . . . . 8 |- ( 2 / 3 ) < 1
28	21, 22, 10	redivclapi 8563	. . . . . . . . 9 |- ( 2 / 3 ) e. RR
29		ltmul2 8638	. . . . . . . . 9 |- ( ( ( 2 / 3 ) e. RR /\ 1 e. RR /\ ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) ) -> ( ( 2 / 3 ) < 1 <-> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) ) )
30	28, 2, 29	mp3an12 1306	. . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( 2 / 3 ) < 1 <-> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) ) )
31	27, 30	mpbii 147	. . . . . . 7 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
32	6, 19, 31	syl2anc 409	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
33	7	mulid1d 7807	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. 1 ) = ( A ^ 2 ) )
34	32, 33	breqtrd 3962	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( A ^ 2 ) )
35	14, 34	eqbrtrd 3958	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) )
36		0re 7790	. . . . . . . . 9 |- 0 e. RR
37		ltle 7875	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
38	36, 37	mpan 421	. . . . . . . 8 |- ( A e. RR -> ( 0 < A -> 0 <_ A ) )
39	38	imdistani 442	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A e. RR /\ 0 <_ A ) )
40		le2sq2 10399	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ A <_ 1 ) ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
41	2, 40	mpanr1 434	. . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
42	39, 41	stoic3 1408	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
43	4, 42	sylbi 120	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
44		sq1 10417	. . . . 5 |- ( 1 ^ 2 ) = 1
45	43, 44	breqtrdi 3977	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ 1 )
46		redivclap 8515	. . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
47	22, 10, 46	mp3an23 1308	. . . . . . 7 |- ( ( A ^ 2 ) e. RR -> ( ( A ^ 2 ) / 3 ) e. RR )
48	6, 47	syl 14	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
49		remulcl 7772	. . . . . 6 |- ( ( 2 e. RR /\ ( ( A ^ 2 ) / 3 ) e. RR ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
50	21, 48, 49	sylancr 411	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
51		ltletr 7877	. . . . . 6 |- ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR /\ ( A ^ 2 ) e. RR /\ 1 e. RR ) -> ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) /\ ( A ^ 2 ) <_ 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 ) )
52	2, 51	mp3an3 1305	. . . . 5 |- ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR /\ ( A ^ 2 ) e. RR ) -> ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) /\ ( A ^ 2 ) <_ 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 ) )
53	50, 6, 52	syl2anc 409	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) /\ ( A ^ 2 ) <_ 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 ) )
54	35, 45, 53	mp2and 430	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 )
55		posdif 8241	. . . 4 |- ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR /\ 1 e. RR ) -> ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 <-> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) ) )
56	50, 2, 55	sylancl 410	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 <-> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) ) )
57	54, 56	mpbid 146	. 2 |- ( A e. ( 0 (,] 1 ) -> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) )
58		cos01bnd 11501	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) /\ ( cos ` A ) < ( 1 - ( ( A ^ 2 ) / 3 ) ) ) )
59	58	simpld 111	. 2 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) )
60		resubcl 8050	. . . 4 |- ( ( 1 e. RR /\ ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
61	2, 50, 60	sylancr 411	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
62	5	recoscld 11467	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( cos ` A ) e. RR )
63		lttr 7862	. . 3 |- ( ( 0 e. RR /\ ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR /\ ( cos ` A ) e. RR ) -> ( ( 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) /\ ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) ) -> 0 < ( cos ` A ) ) )
64	36, 61, 62, 63	mp3an2i 1321	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) /\ ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) ) -> 0 < ( cos ` A ) ) )
65	57, 59, 64	mp2and 430	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( 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   1c1 7645   x. cmul 7649   RR*cxr 7823   < clt 7824   <_ cle 7825   - cmin 7957   # cap 8367   / cdiv 8456   2c2 8795   3c3 8796   ZZcz 9078   (,]cioc 9702   ^cexp 10323   cosccos 11388

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

This theorem depends on definitions:  df-bi 116  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-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-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  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-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ioc 9706  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  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-cos 11394

This theorem is referenced by:  sin02gt0  11506  sincos1sgn  11507  tangtx  12967