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Theorem sin01gt0 7418
Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
sin01gt0 |- (A e. (0(,]1) -> 0 < (sin` A))
Proof of Theorem sin01gt0
StepHypRef Expression
1 0re 5412 . . . . . . . . 9 |- 0 e. RR
2 1re 5407 . . . . . . . . 9 |- 1 e. RR
3 elioc2t 6322 . . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR) -> (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1)))
41, 2, 3mp2an 695 . . . . . . . 8 |- (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1))
54biimp 151 . . . . . . 7 |- (A e. (0(,]1) -> (A e. RR /\ 0 < A /\ A <_ 1))
653simp1d 792 . . . . . 6 |- (A e. (0(,]1) -> A e. RR)
7 3nn 5947 . . . . . . . 8 |- 3 e. NN
87nnnn0 6054 . . . . . . 7 |- 3 e. NN0
9 reexpclt 6512 . . . . . . 7 |- ((A e. RR /\ 3 e. NN0) -> (A^3) e. RR)
108, 9mpan2 694 . . . . . 6 |- (A e. RR -> (A^3) e. RR)
116, 10syl 10 . . . . 5 |- (A e. (0(,]1) -> (A^3) e. RR)
12 3re 5928 . . . . . 6 |- 3 e. RR
137nnne0 5899 . . . . . 6 |- 3 =/= 0
14 redivclt 5756 . . . . . 6 |- (((A^3) e. RR /\ 3 e. RR /\ 3 =/= 0) -> ((A^3) / 3) e. RR)
1512, 13, 14mp3an23 905 . . . . 5 |- ((A^3) e. RR -> ((A^3) / 3) e. RR)
1611, 15syl 10 . . . 4 |- (A e. (0(,]1) -> ((A^3) / 3) e. RR)
17 lt01 5653 . . . . . . . . 9 |- 0 < 1
18 3pos 5938 . . . . . . . . 9 |- 0 < 3
19 2pos 5936 . . . . . . . . . . . . 13 |- 0 < 2
20 2re 5926 . . . . . . . . . . . . . 14 |- 2 e. RR
211, 20, 2ltadd1 5565 . . . . . . . . . . . . 13 |- (0 < 2 <-> (0 + 1) < (2 + 1))
2219, 21mpbi 189 . . . . . . . . . . . 12 |- (0 + 1) < (2 + 1)
23 ax1cn 5241 . . . . . . . . . . . . 13 |- 1 e. CC
2423addid2 5303 . . . . . . . . . . . 12 |- (0 + 1) = 1
25 df-3 5918 . . . . . . . . . . . . 13 |- 3 = (2 + 1)
2625eqcomi 1471 . . . . . . . . . . . 12 |- (2 + 1) = 3
2722, 24, 263brtr3 2632 . . . . . . . . . . 11 |- 1 < 3
28 ltdiv2t 5835 . . . . . . . . . . 11 |- (((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) /\ (0 < 1 /\ 0 < 3 /\ 0 < (A^3))) -> (1 < 3 <-> ((A^3) / 3) < ((A^3) / 1)))
2927, 28mpbii 193 . . . . . . . . . 10 |- (((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) /\ (0 < 1 /\ 0 < 3 /\ 0 < (A^3))) -> ((A^3) / 3) < ((A^3) / 1))
3029expcom 374 . . . . . . . . 9 |- ((0 < 1 /\ 0 < 3 /\ 0 < (A^3)) -> ((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) -> ((A^3) / 3) < ((A^3) / 1)))
3117, 18, 30mp3an12 903 . . . . . . . 8 |- (0 < (A^3) -> ((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) -> ((A^3) / 3) < ((A^3) / 1)))
3231com12 11 . . . . . . 7 |- ((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) -> (0 < (A^3) -> ((A^3) / 3) < ((A^3) / 1)))
332, 12, 32mp3an12 903 . . . . . 6 |- ((A^3) e. RR -> (0 < (A^3) -> ((A^3) / 3) < ((A^3) / 1)))
34 expgt0t 6520 . . . . . . . . 9 |- ((A e. RR /\ 3 e. NN0 /\ 0 < A) -> 0 < (A^3))
358, 34mp3an2 901 . . . . . . . 8 |- ((A e. RR /\ 0 < A) -> 0 < (A^3))
36353adant3 797 . . . . . . 7 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> 0 < (A^3))
374, 36sylbi 199 . . . . . 6 |- (A e. (0(,]1) -> 0 < (A^3))
3833, 11, 37sylc 68 . . . . 5 |- (A e. (0(,]1) -> ((A^3) / 3) < ((A^3) / 1))
3911recnd 5287 . . . . . 6 |- (A e. (0(,]1) -> (A^3) e. CC)
40 div1t 5729 . . . . . 6 |- ((A^3) e. CC -> ((A^3) / 1) = (A^3))
4139, 40syl 10 . . . . 5 |- (A e. (0(,]1) -> ((A^3) / 1) = (A^3))
4238, 41breqtrd 2629 . . . 4 |- (A e. (0(,]1) -> ((A^3) / 3) < (A^3))
43 1nn0 6061 . . . . . . . 8 |- 1 e. NN0
44 expword2it 6536 . . . . . . . . . . 11 |- (((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) /\ (0 < A /\ A <_ 1 /\ 1 < 3)) -> (A^3) <_ (A^1))
4544expcom 374 . . . . . . . . . 10 |- ((0 < A /\ A <_ 1 /\ 1 < 3) -> ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> (A^3) <_ (A^1)))
4627, 45mp3an3 902 . . . . . . . . 9 |- ((0 < A /\ A <_ 1) -> ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> (A^3) <_ (A^1)))
4746com12 11 . . . . . . . 8 |- ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> ((0 < A /\ A <_ 1) -> (A^3) <_ (A^1)))
4843, 8, 47mp3an23 905 . . . . . . 7 |- (A e. RR -> ((0 < A /\ A <_ 1) -> (A^3) <_ (A^1)))
49483impib 829 . . . . . 6 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> (A^3) <_ (A^1))
504, 49sylbi 199 . . . . 5 |- (A e. (0(,]1) -> (A^3) <_ (A^1))
516recnd 5287 . . . . . 6 |- (A e. (0(,]1) -> A e. CC)
52 exp1t 6505 . . . . . 6 |- (A e. CC -> (A^1) = A)
5351, 52syl 10 . . . . 5 |- (A e. (0(,]1) -> (A^1) = A)
5450, 53breqtrd 2629 . . . 4 |- (A e. (0(,]1) -> (A^3) <_ A)
5516, 11, 6, 42, 54ltletrd 5501 . . 3 |- (A e. (0(,]1) -> ((A^3) / 3) < A)
56 posdift 5627 . . . 4 |- ((((A^3) / 3) e. RR /\ A e. RR) -> (((A^3) / 3) < A <-> 0 < (A - ((A^3) / 3))))
5756, 16, 6sylanc 471 . . 3 |- (A e. (0(,]1) -> (((A^3) / 3) < A <-> 0 < (A - ((A^3) / 3))))
5855, 57mpbid 195 . 2 |- (A e. (0(,]1) -> 0 < (A - ((A^3) / 3)))
59 sin01bnd 7414 . . 3 |- (A e. (0(,]1) -> ((A - ((A^3) / 3)) < (sin` A) /\ (sin` A) < A))
6059pm3.26d 321 . 2 |- (A e. (0(,]1) -> (A - ((A^3) / 3)) < (sin` A))
61 axlttrn 5476 . . . 4 |- ((0 e. RR /\ (A - ((A^3) / 3)) e. RR /\ (sin` A) e. RR) -> ((0 < (A - ((A^3) / 3)) /\ (A - ((A^3) / 3)) < (sin` A)) -> 0 < (sin` A)))
621, 61mp3an1 900 . . 3 |- (((A - ((A^3) / 3)) e. RR /\ (sin` A) e. RR) -> ((0 < (A - ((A^3) / 3)) /\ (A - ((A^3) / 3)) < (sin` A)) -> 0 < (sin` A)))
63 resubclt 5410 . . . 4 |- ((A e. RR /\ ((A^3) / 3) e. RR) -> (A - ((A^3) / 3)) e. RR)
6463, 6, 16sylanc 471 . . 3 |- (A e. (0(,]1) -> (A - ((A^3) / 3)) e. RR)
65 resinclt 7380 . . . 4 |- (A e. RR -> (sin` A) e. RR)
666, 65syl 10 . . 3 |- (A e. (0(,]1) -> (sin` A) e. RR)
6762, 64, 66sylanc 471 . 2 |- (A e. (0(,]1) -> ((0 < (A - ((A^3) / 3)) /\ (A - ((A^3) / 3)) < (sin` A)) -> 0 < (sin` A)))
6858, 60, 67mp2and 701 1 |- (A e. (0(,]1) -> 0 < (sin` A))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   + caddc 5209   - cmin 5264   / cdiv 5266   <_ cle 5267  NN0cn0 5269   < clt 5458  2c2 5908  3c3 5909  (,]cioc 6295  ^cexp 6500  sincsin 7237
This theorem is referenced by:  sin02gt0 7420  sincos1sgn 7421  sincos4thpi 8627
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp