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Theorem cos01gt0 14921
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
StepHypRef Expression
1 0xr 10086 . . . . . . . . . 10 |- 0 e. RR*
2 1re 10039 . . . . . . . . . 10 |- 1 e. RR
3 elioc2 12236 . . . . . . . . . 10 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
41, 2, 3mp2an 708 . . . . . . . . 9 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
54simp1bi 1076 . . . . . . . 8 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
65resqcld 13035 . . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. RR )
76recnd 10068 . . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) e. CC )
8 2cn 11091 . . . . . . 7 |- 2 e. CC
9 3cn 11095 . . . . . . . 8 |- 3 e. CC
10 3ne0 11115 . . . . . . . 8 |- 3 =/= 0
119, 10pm3.2i 471 . . . . . . 7 |- ( 3 e. CC /\ 3 =/= 0 )
12 div12 10707 . . . . . . 7 |- ( ( 2 e. CC /\ ( A ^ 2 ) e. CC /\ ( 3 e. CC /\ 3 =/= 0 ) ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
138, 11, 12mp3an13 1415 . . . . . 6 |- ( ( A ^ 2 ) e. CC -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
147, 13syl 17 . . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) = ( ( A ^ 2 ) x. ( 2 / 3 ) ) )
15 2z 11409 . . . . . . . . . 10 |- 2 e. ZZ
16 expgt0 12893 . . . . . . . . . 10 |- ( ( A e. RR /\ 2 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 2 ) )
1715, 16mp3an2 1412 . . . . . . . . 9 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 2 ) )
18173adant3 1081 . . . . . . . 8 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> 0 < ( A ^ 2 ) )
194, 18sylbi 207 . . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A ^ 2 ) )
20 2lt3 11195 . . . . . . . . . 10 |- 2 < 3
21 2re 11090 . . . . . . . . . . 11 |- 2 e. RR
22 3re 11094 . . . . . . . . . . 11 |- 3 e. RR
23 3pos 11114 . . . . . . . . . . 11 |- 0 < 3
2421, 22, 22, 23ltdiv1ii 10953 . . . . . . . . . 10 |- ( 2 < 3 <-> ( 2 / 3 ) < ( 3 / 3 ) )
2520, 24mpbi 220 . . . . . . . . 9 |- ( 2 / 3 ) < ( 3 / 3 )
269, 10dividi 10758 . . . . . . . . 9 |- ( 3 / 3 ) = 1
2725, 26breqtri 4678 . . . . . . . 8 |- ( 2 / 3 ) < 1
2821, 22, 10redivcli 10792 . . . . . . . . 9 |- ( 2 / 3 ) e. RR
29 ltmul2 10874 . . . . . . . . 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 ) ) )
3028, 2, 29mp3an12 1414 . . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( 2 / 3 ) < 1 <-> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) ) )
3127, 30mpbii 223 . . . . . . 7 |- ( ( ( A ^ 2 ) e. RR /\ 0 < ( A ^ 2 ) ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
326, 19, 31syl2anc 693 . . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( ( A ^ 2 ) x. 1 ) )
337mulid1d 10057 . . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. 1 ) = ( A ^ 2 ) )
3432, 33breqtrd 4679 . . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) x. ( 2 / 3 ) ) < ( A ^ 2 ) )
3514, 34eqbrtrd 4675 . . . 4 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) )
36 0re 10040 . . . . . . . . 9 |- 0 e. RR
37 ltle 10126 . . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
3836, 37mpan 706 . . . . . . . 8 |- ( A e. RR -> ( 0 < A -> 0 <_ A ) )
3938imdistani 726 . . . . . . 7 |- ( ( A e. RR /\ 0 < A ) -> ( A e. RR /\ 0 <_ A ) )
40 le2sq2 12939 . . . . . . . 8 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( 1 e. RR /\ A <_ 1 ) ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
412, 40mpanr1 719 . . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
4239, 41stoic3 1701 . . . . . 6 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
434, 42sylbi 207 . . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ ( 1 ^ 2 ) )
44 sq1 12958 . . . . 5 |- ( 1 ^ 2 ) = 1
4543, 44syl6breq 4694 . . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 2 ) <_ 1 )
46 redivcl 10744 . . . . . . . 8 |- ( ( ( A ^ 2 ) e. RR /\ 3 e. RR /\ 3 =/= 0 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
4722, 10, 46mp3an23 1416 . . . . . . 7 |- ( ( A ^ 2 ) e. RR -> ( ( A ^ 2 ) / 3 ) e. RR )
486, 47syl 17 . . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 2 ) / 3 ) e. RR )
49 remulcl 10021 . . . . . 6 |- ( ( 2 e. RR /\ ( ( A ^ 2 ) / 3 ) e. RR ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
5021, 48, 49sylancr 695 . . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR )
51 ltletr 10129 . . . . . 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 ) )
522, 51mp3an3 1413 . . . . 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 ) )
5350, 6, 52syl2anc 693 . . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < ( A ^ 2 ) /\ ( A ^ 2 ) <_ 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 ) )
5435, 45, 53mp2and 715 . . 3 |- ( A e. ( 0 (,] 1 ) -> ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 )
55 posdif 10521 . . . 4 |- ( ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR /\ 1 e. RR ) -> ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 <-> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) ) )
5650, 2, 55sylancl 694 . . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 2 x. ( ( A ^ 2 ) / 3 ) ) < 1 <-> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) ) )
5754, 56mpbid 222 . 2 |- ( A e. ( 0 (,] 1 ) -> 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) )
58 cos01bnd 14916 . . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) /\ ( cos ` A ) < ( 1 - ( ( A ^ 2 ) / 3 ) ) ) )
5958simpld 475 . 2 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) )
60 resubcl 10345 . . . 4 |- ( ( 1 e. RR /\ ( 2 x. ( ( A ^ 2 ) / 3 ) ) e. RR ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
612, 50, 60sylancr 695 . . 3 |- ( A e. ( 0 (,] 1 ) -> ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) e. RR )
625recoscld 14874 . . 3 |- ( A e. ( 0 (,] 1 ) -> ( cos ` A ) e. RR )
63 lttr 10114 . . . 4 |- ( ( 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 ) ) )
6436, 63mp3an1 1411 . . 3 |- ( ( ( 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 ) ) )
6561, 62, 64syl2anc 693 . 2 |- ( A e. ( 0 (,] 1 ) -> ( ( 0 < ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) /\ ( 1 - ( 2 x. ( ( A ^ 2 ) / 3 ) ) ) < ( cos ` A ) ) -> 0 < ( cos ` A ) ) )
6657, 59, 65mp2and 715 1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( cos ` A ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   3c3 11071   ZZcz 11377   (,]cioc 12176   ^cexp 12860   cosccos 14795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioc 12180  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-cos 14801
This theorem is referenced by:  sin02gt0  14922  sincos1sgn  14923  tangtx  24257
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