Theorem sin01gt0 11927

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 8073	. . . . . . . 8 |- 0 e. RR*
2		1re 8025	. . . . . . . 8 |- 1 e. RR
3		elioc2 10011	. . . . . . . 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 1014	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6		3nn0 9267	. . . . . 6 |- 3 e. NN0
7		reexpcl 10648	. . . . . 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 9064	. . . . . 6 |- 3 e. RR
10		3ap0 9086	. . . . . 6 |- 3 # 0
11		redivclap 8758	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
12	9, 10, 11	mp3an23 1340	. . . . 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 9355	. . . . . . . . 9 |- 3 e. ZZ
15		expgt0 10664	. . . . . . . . 9 |- ( ( A e. RR /\ 3 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 3 ) )
16	14, 15	mp3an2 1336	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 3 ) )
17	16	3adant3 1019	. . . . . . 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 8153	. . . . . . . 8 |- 0 < 1
20	2, 19	pm3.2i 272	. . . . . . 7 |- ( 1 e. RR /\ 0 < 1 )
21		3pos 9084	. . . . . . . 8 |- 0 < 3
22	9, 21	pm3.2i 272	. . . . . . 7 |- ( 3 e. RR /\ 0 < 3 )
23		1lt3 9162	. . . . . . . 8 |- 1 < 3
24		ltdiv2 8914	. . . . . . . 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 1338	. . . . . 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 8055	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. CC )
29	28	div1d 8807	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 1 ) = ( A ^ 3 ) )
30	27, 29	breqtrd 4059	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( A ^ 3 ) )
31		1nn0 9265	. . . . . . 7 |- 1 e. NN0
32	31	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 1 e. NN0 )
33		1le3 9202	. . . . . . . 8 |- 1 <_ 3
34		1z 9352	. . . . . . . . 9 |- 1 e. ZZ
35	34	eluz1i 9608	. . . . . . . 8 |- ( 3 e. ( ZZ>= ` 1 ) <-> ( 3 e. ZZ /\ 1 <_ 3 ) )
36	14, 33, 35	mpbir2an 944	. . . . . . 7 |- 3 e. ( ZZ>= ` 1 )
37	36	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 3 e. ( ZZ>= ` 1 ) )
38	4	simp2bi 1015	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
39		0re 8026	. . . . . . . 8 |- 0 e. RR
40		ltle 8114	. . . . . . . 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 1016	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
44	5, 32, 37, 42, 43	leexp2rd 10795	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ ( A ^ 1 ) )
45	5	recnd 8055	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
46	45	exp1d 10760	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 1 ) = A )
47	44, 46	breqtrd 4059	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ A )
48	13, 8, 5, 30, 47	ltletrd 8450	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < A )
49	13, 5	posdifd 8559	. . 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 11922	. . 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 8407	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) e. RR )
54	5	resincld 11888	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( sin ` A ) e. RR )
55		lttr 8100	. . 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 1353	. 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 980   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   RR*cxr 8060   < clt 8061   <_ cle 8062   - cmin 8197   # cap 8608   / cdiv 8699   3c3 9042   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   (,]cioc 9964   ^cexp 10630   sincsin 11809

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-ioc 9968  df-ico 9969  df-fz 10084  df-fzo 10218  df-seqfrec 10540  df-exp 10631  df-fac 10818  df-ihash 10868  df-shft 10980  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-clim 11444  df-sumdc 11519  df-ef 11813  df-sin 11815

This theorem is referenced by:  sin02gt0  11929  sincos1sgn  11930  sincos4thpi  15076