Theorem sin01gt0 7435

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 5423	. . . . . . . . 9 |- 0 e. RR
2		1re 5418	. . . . . . . . 9 |- 1 e. RR
3		elioc2t 6335	. . . . . . . . 9 |- ((0 e. RR /\ 1 e. RR) -> (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1)))
4	1, 2, 3	mp2an 696	. . . . . . . 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 793	. . . . . 6 |- (A e. (0(,]1) -> A e. RR)
7		3nn 5957	. . . . . . . 8 |- 3 e. NN
8	7	nnnn0 6064	. . . . . . 7 |- 3 e. NN0
9		reexpclt 6525	. . . . . . 7 |- ((A e. RR /\ 3 e. NN0) -> (A^3) e. RR)
10	8, 9	mpan2 695	. . . . . 6 |- (A e. RR -> (A^3) e. RR)
11	6, 10	syl 10	. . . . 5 |- (A e. (0(,]1) -> (A^3) e. RR)
12		3re 5938	. . . . . 6 |- 3 e. RR
13	7	nnne0 5909	. . . . . 6 |- 3 =/= 0
14		redivclt 5766	. . . . . 6 |- (((A^3) e. RR /\ 3 e. RR /\ 3 =/= 0) -> ((A^3) / 3) e. RR)
15	12, 13, 14	mp3an23 907	. . . . 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 5663	. . . . . . . . 9 |- 0 < 1
18		3pos 5948	. . . . . . . . 9 |- 0 < 3
19		2pos 5946	. . . . . . . . . . . . 13 |- 0 < 2
20		2re 5936	. . . . . . . . . . . . . 14 |- 2 e. RR
21	1, 20, 2	ltadd1 5575	. . . . . . . . . . . . 13 |- (0 < 2 <-> (0 + 1) < (2 + 1))
22	19, 21	mpbi 189	. . . . . . . . . . . 12 |- (0 + 1) < (2 + 1)
23		ax1cn 5252	. . . . . . . . . . . . 13 |- 1 e. CC
24	23	addid2 5314	. . . . . . . . . . . 12 |- (0 + 1) = 1
25		df-3 5928	. . . . . . . . . . . . 13 |- 3 = (2 + 1)
26	25	eqcomi 1477	. . . . . . . . . . . 12 |- (2 + 1) = 3
27	22, 24, 26	3brtr3 2638	. . . . . . . . . . 11 |- 1 < 3
28		ltdiv2t 5845	. . . . . . . . . . 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 905	. . . . . . . 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 905	. . . . . 6 |- ((A^3) e. RR -> (0 < (A^3) -> ((A^3) / 3) < ((A^3) / 1)))
34		expgt0t 6534	. . . . . . . . 9 |- ((A e. RR /\ 3 e. NN0 /\ 0 < A) -> 0 < (A^3))
35	8, 34	mp3an2 903	. . . . . . . 8 |- ((A e. RR /\ 0 < A) -> 0 < (A^3))
36	35	3adant3 798	. . . . . . 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 5298	. . . . . 6 |- (A e. (0(,]1) -> (A^3) e. CC)
40		div1t 5739	. . . . . 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 2635	. . . 4 |- (A e. (0(,]1) -> ((A^3) / 3) < (A^3))
43		1nn0 6071	. . . . . . . 8 |- 1 e. NN0
44		expword2it 6550	. . . . . . . . . . 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 904	. . . . . . . . 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 907	. . . . . . 7 |- (A e. RR -> ((0 < A /\ A <_ 1) -> (A^3) <_ (A^1)))
49	48	3impib 830	. . . . . 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 5298	. . . . . 6 |- (A e. (0(,]1) -> A e. CC)
52		exp1t 6518	. . . . . 6 |- (A e. CC -> (A^1) = A)
53	51, 52	syl 10	. . . . 5 |- (A e. (0(,]1) -> (A^1) = A)
54	50, 53	breqtrd 2635	. . . 4 |- (A e. (0(,]1) -> (A^3) <_ A)
55	16, 11, 6, 42, 54	ltletrd 5511	. . 3 |- (A e. (0(,]1) -> ((A^3) / 3) < A)
56		posdift 5637	. . . 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 7431	. . 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 5487	. . . 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 902	. . 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 5421	. . . 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 7397	. . . 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 702	1 |- (A e. (0(,]1) -> 0 < (sin` A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957   =/= wne 1583   class class class wbr 2615  ` cfv 3178  (class class class)co 3958  CCcc 5215  RRcr 5216  0cc0 5217  1c1 5218   + caddc 5220   - cmin 5275   / cdiv 5277   <_ cle 5278  NN0cn0 5280   < clt 5469  2c2 5918  3c3 5919  (,]cioc 6308  ^cexp 6513  sincsin 7254

This theorem is referenced by:  sin02gt0 7437  sincos1sgn 7438  sincos4thpi 8662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-sup 4557  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp