Theorem cos01gt0 11725

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 7966	. . . . . . . . . 10 |- 0 e. RR*
2		1re 7919	. . . . . . . . . 10 |- 1 e. RR
3		elioc2 9893	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 424	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 1007	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6	5	resqcld 10635	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. RR )
7	6	recnd 7948	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. CC )
8		2cn 8949	. . . . . . 7 |- 2 e. CC
9		3cn 8953	. . . . . . . 8 |- 3 e. CC
10		3ap0 8974	. . . . . . . 8 |- 3 # 0
11	9, 10	pm3.2i 270	. . . . . . 7 |- ( 3 e. CC /\ 3 # 0 )
12		div12ap 8611	. . . . . . 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 1323	. . . . . 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 9240	. . . . . . . . . 10 |- 2 e. ZZ
16		expgt0 10509	. . . . . . . . . 10 |- ( ( A e. RR /\ 2 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 2 ) )
17	15, 16	mp3an2 1320	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 2 ) )
18	17	3adant3 1012	. . . . . . . 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 9048	. . . . . . . . . 10 |- 2 < 3
21		2re 8948	. . . . . . . . . . 11 |- 2 e. RR
22		3re 8952	. . . . . . . . . . 11 |- 3 e. RR
23		3pos 8972	. . . . . . . . . . 11 |- 0 < 3
24	21, 22, 22, 23	ltdiv1ii 8845	. . . . . . . . . 10 |- ( 2 < 3 <-> ( 2 / 3 ) < ( 3 / 3 ) )
25	20, 24	mpbi 144	. . . . . . . . 9 |- ( 2 / 3 ) < ( 3 / 3 )
26	9, 10	dividapi 8662	. . . . . . . . 9 |- ( 3 / 3 ) = 1
27	25, 26	breqtri 4014	. . . . . . . 8 |- ( 2 / 3 ) < 1
28	21, 22, 10	redivclapi 8696	. . . . . . . . 9 |- ( 2 / 3 ) e. RR
29		ltmul2 8772	. . . . . . . . 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 1322	. . . . . . . 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 7937	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. 1 ) = ( A ^ 2 ) )
34	32, 33	breqtrd 4015	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( A ^ 2 ) )
35	14, 34	eqbrtrd 4011	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) )
36		0re 7920	. . . . . . . . 9 |- 0 e. RR
37		ltle 8007	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
38	36, 37	mpan 422	. . . . . . . 8 |- ( A e. RR -> ( 0 < A -> 0 <_ A ) )
39	38	imdistani 443	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A e. RR /\ 0 <_ A ) )
40		le2sq2 10551	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ A <_ 1 ) ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
41	2, 40	mpanr1 435	. . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
42	39, 41	stoic3 1424	. . . . . 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 10569	. . . . 5 |- ( 1 ^ 2 ) = 1
45	43, 44	breqtrdi 4030	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ 1 )
46		redivclap 8648	. . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
47	22, 10, 46	mp3an23 1324	. . . . . . 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 7902	. . . . . 6 |- ( ( 2 e. RR /\ ( ( A ^ 2 ) / 3 ) e. RR ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
50	21, 48, 49	sylancr 412	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
51		ltletr 8009	. . . . . 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 1321	. . . . 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 431	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 )
55		posdif 8374	. . . 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 411	. . 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 11721	. . 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 8183	. . . 4 |- ( ( 1 e. RR /\ ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
61	2, 50, 60	sylancr 412	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
62	5	recoscld 11687	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( cos ` A ) e. RR )
63		lttr 7993	. . 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 1337	. 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 431	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( cos ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   x. cmul 7779   RR*cxr 7953   < clt 7954   <_ cle 7955   - cmin 8090   # cap 8500   / cdiv 8589   2c2 8929   3c3 8930   ZZcz 9212   (,]cioc 9846   ^cexp 10475   cosccos 11608

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ioc 9850  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-cos 11614

This theorem is referenced by:  sin02gt0  11726  sincos1sgn  11727  tangtx  13553