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

Step	Hyp	Ref	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)))
4	1, 2, 3	mp2an 695	. . . . . . . 8 |- (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1))
5	4	biimp 151	. . . . . . 7 |- (A e. (0(,]1) -> (A e. RR /\ 0 < A /\ A <_ 1))
6	5	3simp1d 792	. . . . . 6 |- (A e. (0(,]1) -> A e. RR)
7		3nn 5947	. . . . . . . 8 |- 3 e. NN
8	7	nnnn0 6054	. . . . . . 7 |- 3 e. NN0
9		reexpclt 6512	. . . . . . 7 |- ((A e. RR /\ 3 e. NN0) -> (A^3) e. RR)
10	8, 9	mpan2 694	. . . . . 6 |- (A e. RR -> (A^3) e. RR)
11	6, 10	syl 10	. . . . 5 |- (A e. (0(,]1) -> (A^3) e. RR)
12		3re 5928	. . . . . 6 |- 3 e. RR
13	7	nnne0 5899	. . . . . 6 |- 3 =/= 0
14		redivclt 5756	. . . . . 6 |- (((A^3) e. RR /\ 3 e. RR /\ 3 =/= 0) -> ((A^3) / 3) e. RR)
15	12, 13, 14	mp3an23 905	. . . . 5 |- ((A^3) e. RR -> ((A^3) / 3) e. RR)
16	11, 15	syl 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
21	1, 20, 2	ltadd1 5565	. . . . . . . . . . . . 13 |- (0 < 2 <-> (0 + 1) < (2 + 1))
22	19, 21	mpbi 189	. . . . . . . . . . . 12 |- (0 + 1) < (2 + 1)
23		ax1cn 5241	. . . . . . . . . . . . 13 |- 1 e. CC
24	23	addid2 5303	. . . . . . . . . . . 12 |- (0 + 1) = 1
25		df-3 5918	. . . . . . . . . . . . 13 |- 3 = (2 + 1)
26	25	eqcomi 1471	. . . . . . . . . . . 12 |- (2 + 1) = 3
27	22, 24, 26	3brtr3 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)))
29	27, 28	mpbii 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))
30	29	expcom 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)))
31	17, 18, 30	mp3an12 903	. . . . . . . 8 |- (0 < (A^3) -> ((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) -> ((A^3) / 3) < ((A^3) / 1)))
32	31	com12 11	. . . . . . 7 |- ((1 e. RR /\ 3 e. RR /\ (A^3) e. RR) -> (0 < (A^3) -> ((A^3) / 3) < ((A^3) / 1)))
33	2, 12, 32	mp3an12 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))
35	8, 34	mp3an2 901	. . . . . . . 8 |- ((A e. RR /\ 0 < A) -> 0 < (A^3))
36	35	3adant3 797	. . . . . . 7 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> 0 < (A^3))
37	4, 36	sylbi 199	. . . . . 6 |- (A e. (0(,]1) -> 0 < (A^3))
38	33, 11, 37	sylc 68	. . . . 5 |- (A e. (0(,]1) -> ((A^3) / 3) < ((A^3) / 1))
39	11	recnd 5287	. . . . . 6 |- (A e. (0(,]1) -> (A^3) e. CC)
40		div1t 5729	. . . . . 6 |- ((A^3) e. CC -> ((A^3) / 1) = (A^3))
41	39, 40	syl 10	. . . . 5 |- (A e. (0(,]1) -> ((A^3) / 1) = (A^3))
42	38, 41	breqtrd 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))
45	44	expcom 374	. . . . . . . . . 10 |- ((0 < A /\ A <_ 1 /\ 1 < 3) -> ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> (A^3) <_ (A^1)))
46	27, 45	mp3an3 902	. . . . . . . . 9 |- ((0 < A /\ A <_ 1) -> ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> (A^3) <_ (A^1)))
47	46	com12 11	. . . . . . . 8 |- ((A e. RR /\ 1 e. NN0 /\ 3 e. NN0) -> ((0 < A /\ A <_ 1) -> (A^3) <_ (A^1)))
48	43, 8, 47	mp3an23 905	. . . . . . 7 |- (A e. RR -> ((0 < A /\ A <_ 1) -> (A^3) <_ (A^1)))
49	48	3impib 829	. . . . . 6 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> (A^3) <_ (A^1))
50	4, 49	sylbi 199	. . . . 5 |- (A e. (0(,]1) -> (A^3) <_ (A^1))
51	6	recnd 5287	. . . . . 6 |- (A e. (0(,]1) -> A e. CC)
52		exp1t 6505	. . . . . 6 |- (A e. CC -> (A^1) = A)
53	51, 52	syl 10	. . . . 5 |- (A e. (0(,]1) -> (A^1) = A)
54	50, 53	breqtrd 2629	. . . 4 |- (A e. (0(,]1) -> (A^3) <_ A)
55	16, 11, 6, 42, 54	ltletrd 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))))
57	56, 16, 6	sylanc 471	. . 3 |- (A e. (0(,]1) -> (((A^3) / 3) < A <-> 0 < (A - ((A^3) / 3))))
58	55, 57	mpbid 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))
60	59	pm3.26d 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)))
62	1, 61	mp3an1 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)
64	63, 6, 16	sylanc 471	. . 3 |- (A e. (0(,]1) -> (A - ((A^3) / 3)) e. RR)
65		resinclt 7380	. . . 4 |- (A e. RR -> (sin` A) e. RR)
66	6, 65	syl 10	. . 3 |- (A e. (0(,]1) -> (sin` A) e. RR)
67	62, 64, 66	sylanc 471	. 2 |- (A e. (0(,]1) -> ((0 < (A - ((A^3) / 3)) /\ (A - ((A^3) / 3)) < (sin` A)) -> 0 < (sin` A)))
68	58, 60, 67	mp2and 701	1 |- (A e. (0(,]1) -> 0 < (sin` A))

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