Theorem sin01gt0 11771

Description: The sine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Wolf Lammen, 25-Sep-2020.)

Assertion

Ref	Expression
sin01gt0	|- ( A e. ( 0 (,] 1 ) -> 0 < ( sin ` A ) )

Proof of Theorem sin01gt0

Step	Hyp	Ref	Expression
1		0xr 8006	. . . . . . . 8 |- 0 e. RR*
2		1re 7958	. . . . . . . 8 |- 1 e. RR
3		elioc2 9938	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 426	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 1012	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6		3nn0 9196	. . . . . 6 |- 3 e. NN0
7		reexpcl 10539	. . . . . 6 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
8	5, 6, 7	sylancl 413	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. RR )
9		3re 8995	. . . . . 6 |- 3 e. RR
10		3ap0 9017	. . . . . 6 |- 3 # 0
11		redivclap 8690	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
12	9, 10, 11	mp3an23 1329	. . . . 5 |- ( ( A ^ 3 ) e. RR -> ( ( A ^ 3 ) / 3 ) e. RR )
13	8, 12	syl 14	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
14		3z 9284	. . . . . . . . 9 |- 3 e. ZZ
15		expgt0 10555	. . . . . . . . 9 |- ( ( A e. RR /\ 3 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 3 ) )
16	14, 15	mp3an2 1325	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 3 ) )
17	16	3adant3 1017	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> 0 < ( A ^ 3 ) )
18	4, 17	sylbi 121	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A ^ 3 ) )
19		0lt1 8086	. . . . . . . 8 |- 0 < 1
20	2, 19	pm3.2i 272	. . . . . . 7 |- ( 1 e. RR /\ 0 < 1 )
21		3pos 9015	. . . . . . . 8 |- 0 < 3
22	9, 21	pm3.2i 272	. . . . . . 7 |- ( 3 e. RR /\ 0 < 3 )
23		1lt3 9092	. . . . . . . 8 |- 1 < 3
24		ltdiv2 8846	. . . . . . . 8 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( 3 e. RR /\ 0 < 3 ) /\ ( ( A ^ 3 ) e. RR /\ 0 < ( A ^ 3 ) ) ) -> ( 1 < 3 <-> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) ) )
25	23, 24	mpbii 148	. . . . . . 7 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( 3 e. RR /\ 0 < 3 ) /\ ( ( A ^ 3 ) e. RR /\ 0 < ( A ^ 3 ) ) ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
26	20, 22, 25	mp3an12 1327	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 0 < ( A ^ 3 ) ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
27	8, 18, 26	syl2anc 411	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
28	8	recnd 7988	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. CC )
29	28	div1d 8739	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 1 ) = ( A ^ 3 ) )
30	27, 29	breqtrd 4031	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( A ^ 3 ) )
31		1nn0 9194	. . . . . . 7 |- 1 e. NN0
32	31	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 1 e. NN0 )
33		1le3 9132	. . . . . . . 8 |- 1 <_ 3
34		1z 9281	. . . . . . . . 9 |- 1 e. ZZ
35	34	eluz1i 9537	. . . . . . . 8 |- ( 3 e. ( ZZ>= ` 1 ) <-> ( 3 e. ZZ /\ 1 <_ 3 ) )
36	14, 33, 35	mpbir2an 942	. . . . . . 7 |- 3 e. ( ZZ>= ` 1 )
37	36	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 3 e. ( ZZ>= ` 1 ) )
38	4	simp2bi 1013	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
39		0re 7959	. . . . . . . 8 |- 0 e. RR
40		ltle 8047	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
41	39, 5, 40	sylancr 414	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 0 < A -> 0 <_ A ) )
42	38, 41	mpd 13	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 0 <_ A )
43	4	simp3bi 1014	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
44	5, 32, 37, 42, 43	leexp2rd 10686	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ ( A ^ 1 ) )
45	5	recnd 7988	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
46	45	exp1d 10651	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 1 ) = A )
47	44, 46	breqtrd 4031	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ A )
48	13, 8, 5, 30, 47	ltletrd 8382	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < A )
49	13, 5	posdifd 8491	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( ( A ^ 3 ) / 3 ) < A <-> 0 < ( A - ( ( A ^ 3 ) / 3 ) ) ) )
50	48, 49	mpbid 147	. 2 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A - ( ( A ^ 3 ) / 3 ) ) )
51		sin01bnd 11767	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) /\ ( sin ` A ) < A ) )
52	51	simpld 112	. 2 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) )
53	5, 13	resubcld 8340	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) e. RR )
54	5	resincld 11733	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( sin ` A ) e. RR )
55		lttr 8033	. . 3 |- ( ( 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 ) ) )
56	39, 53, 54, 55	mp3an2i 1342	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( 0 < ( A - ( ( A ^ 3 ) / 3 ) ) /\ ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) ) -> 0 < ( sin ` A ) ) )
57	50, 52, 56	mp2and 433	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814   RR*cxr 7993   < clt 7994   <_ cle 7995   - cmin 8130   # cap 8540   / cdiv 8631   3c3 8973   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   (,]cioc 9891   ^cexp 10521   sincsin 11654

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-5 8983  df-6 8984  df-7 8985  df-8 8986  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ioc 9895  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-ihash 10758  df-shft 10826  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364  df-ef 11658  df-sin 11660

This theorem is referenced by:  sin02gt0  11773  sincos1sgn  11774  sincos4thpi  14346