Theorem cos01gt0 12107

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 8121	. . . . . . . . . 10 |- 0 e. RR*
2		1re 8073	. . . . . . . . . 10 |- 1 e. RR
3		elioc2 10060	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 426	. . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 1015	. . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6	5	resqcld 10846	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. RR )
7	6	recnd 8103	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. CC )
8		2cn 9109	. . . . . . 7 |- 2 e. CC
9		3cn 9113	. . . . . . . 8 |- 3 e. CC
10		3ap0 9134	. . . . . . . 8 |- 3 # 0
11	9, 10	pm3.2i 272	. . . . . . 7 |- ( 3 e. CC /\ 3 # 0 )
12		div12ap 8769	. . . . . . 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 1341	. . . . . 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 9402	. . . . . . . . . 10 |- 2 e. ZZ
16		expgt0 10719	. . . . . . . . . 10 |- ( ( A e. RR /\ 2 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 2 ) )
17	15, 16	mp3an2 1338	. . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 2 ) )
18	17	3adant3 1020	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> 0 < ( A ^ 2 ) )
19	4, 18	sylbi 121	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A ^ 2 ) )
20		2lt3 9209	. . . . . . . . . 10 |- 2 < 3
21		2re 9108	. . . . . . . . . . 11 |- 2 e. RR
22		3re 9112	. . . . . . . . . . 11 |- 3 e. RR
23		3pos 9132	. . . . . . . . . . 11 |- 0 < 3
24	21, 22, 22, 23	ltdiv1ii 9004	. . . . . . . . . 10 |- ( 2 < 3 <-> ( 2 / 3 ) < ( 3 / 3 ) )
25	20, 24	mpbi 145	. . . . . . . . 9 |- ( 2 / 3 ) < ( 3 / 3 )
26	9, 10	dividapi 8820	. . . . . . . . 9 |- ( 3 / 3 ) = 1
27	25, 26	breqtri 4070	. . . . . . . 8 |- ( 2 / 3 ) < 1
28	21, 22, 10	redivclapi 8854	. . . . . . . . 9 |- ( 2 / 3 ) e. RR
29		ltmul2 8931	. . . . . . . . 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 1340	. . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( 2 / 3 ) < 1 <-> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) ) )
31	27, 30	mpbii 148	. . . . . . 7 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
32	6, 19, 31	syl2anc 411	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
33	7	mulridd 8091	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. 1 ) = ( A ^ 2 ) )
34	32, 33	breqtrd 4071	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( A ^ 2 ) )
35	14, 34	eqbrtrd 4067	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) )
36		0re 8074	. . . . . . . . 9 |- 0 e. RR
37		ltle 8162	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
38	36, 37	mpan 424	. . . . . . . 8 |- ( A e. RR -> ( 0 < A -> 0 <_ A ) )
39	38	imdistani 445	. . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A e. RR /\ 0 <_ A ) )
40		le2sq2 10762	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ A <_ 1 ) ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
41	2, 40	mpanr1 437	. . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
42	39, 41	stoic3 1451	. . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
43	4, 42	sylbi 121	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
44		sq1 10780	. . . . 5 |- ( 1 ^ 2 ) = 1
45	43, 44	breqtrdi 4086	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ 1 )
46		redivclap 8806	. . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
47	22, 10, 46	mp3an23 1342	. . . . . . 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 8055	. . . . . 6 |- ( ( 2 e. RR /\ ( ( A ^ 2 ) / 3 ) e. RR ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
50	21, 48, 49	sylancr 414	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
51		ltletr 8164	. . . . . 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 1339	. . . . 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 411	. . . 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 433	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 )
55		posdif 8530	. . . 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 413	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 <-> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) ) )
57	54, 56	mpbid 147	. 2 |- ( A e. ( 0 (,] 1 ) -> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) )
58		cos01bnd 12102	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) /\ ( cos ` A ) < ( 1 - ( ( A ^ 2 ) / 3 ) ) ) )
59	58	simpld 112	. 2 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) )
60		resubcl 8338	. . . 4 |- ( ( 1 e. RR /\ ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
61	2, 50, 60	sylancr 414	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
62	5	recoscld 12068	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( cos ` A ) e. RR )
63		lttr 8148	. . 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 1355	. 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 433	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( cos ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   CCcc 7925   RRcr 7926   0cc0 7927   1c1 7928   x. cmul 7932   RR*cxr 8108   < clt 8109   <_ cle 8110   - cmin 8245   # cap 8656   / cdiv 8747   2c2 9089   3c3 9090   ZZcz 9374   (,]cioc 10013   ^cexp 10685   cosccos 11989

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 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047

This theorem depends on definitions:  df-bi 117  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-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-5 9100  df-6 9101  df-7 9102  df-8 9103  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-ioc 10017  df-ico 10018  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-ihash 10923  df-shft 11159  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992  df-cos 11995

This theorem is referenced by:  sin02gt0  12108  sincos1sgn  12109  tangtx  15343