Theorem sin01gt0 14920

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 10086	. . . . . . . 8 |- 0 e. RR*
2		1re 10039	. . . . . . . 8 |- 1 e. RR
3		elioc2 12236	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 708	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 1076	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6		3nn0 11310	. . . . . 6 |- 3 e. NN0
7		reexpcl 12877	. . . . . 6 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
8	5, 6, 7	sylancl 694	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. RR )
9		3re 11094	. . . . . 6 |- 3 e. RR
10		3ne0 11115	. . . . . 6 |- 3 =/= 0
11		redivcl 10744	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 3 e. RR /\ 3 =/= 0 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
12	9, 10, 11	mp3an23 1416	. . . . 5 |- ( ( A ^ 3 ) e. RR -> ( ( A ^ 3 ) / 3 ) e. RR )
13	8, 12	syl 17	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
14		3z 11410	. . . . . . . . 9 |- 3 e. ZZ
15		expgt0 12893	. . . . . . . . 9 |- ( ( A e. RR /\ 3 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 3 ) )
16	14, 15	mp3an2 1412	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 3 ) )
17	16	3adant3 1081	. . . . . . 7 |- ( ( A e. RR /\ 0 < A /\ A <_ 1 ) -> 0 < ( A ^ 3 ) )
18	4, 17	sylbi 207	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A ^ 3 ) )
19		0lt1 10550	. . . . . . . . 9 |- 0 < 1
20	2, 19	pm3.2i 471	. . . . . . . 8 |- ( 1 e. RR /\ 0 < 1 )
21		3pos 11114	. . . . . . . . 9 |- 0 < 3
22	9, 21	pm3.2i 471	. . . . . . . 8 |- ( 3 e. RR /\ 0 < 3 )
23		1lt3 11196	. . . . . . . . 9 |- 1 < 3
24		ltdiv2 10909	. . . . . . . . 9 |- ( ( ( 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 223	. . . . . . . 8 |- ( ( ( 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 1414	. . . . . . 7 |- ( ( ( A ^ 3 ) e. RR /\ 0 < ( A ^ 3 ) ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
27	26	ex 450	. . . . . 6 |- ( ( A ^ 3 ) e. RR -> ( 0 < ( A ^ 3 ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) ) )
28	8, 18, 27	sylc 65	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
29	8	recnd 10068	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. CC )
30	29	div1d 10793	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 1 ) = ( A ^ 3 ) )
31	28, 30	breqtrd 4679	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( A ^ 3 ) )
32		1nn0 11308	. . . . . . 7 |- 1 e. NN0
33	32	a1i 11	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 1 e. NN0 )
34		1le3 11244	. . . . . . . 8 |- 1 <_ 3
35		1z 11407	. . . . . . . . 9 |- 1 e. ZZ
36	35	eluz1i 11695	. . . . . . . 8 |- ( 3 e. ( ZZ>= ` 1 ) <-> ( 3 e. ZZ /\ 1 <_ 3 ) )
37	14, 34, 36	mpbir2an 955	. . . . . . 7 |- 3 e. ( ZZ>= ` 1 )
38	37	a1i 11	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 3 e. ( ZZ>= ` 1 ) )
39	4	simp2bi 1077	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
40		0re 10040	. . . . . . . 8 |- 0 e. RR
41		ltle 10126	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
42	40, 5, 41	sylancr 695	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 0 < A -> 0 <_ A ) )
43	39, 42	mpd 15	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 0 <_ A )
44	4	simp3bi 1078	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
45	5, 33, 38, 43, 44	leexp2rd 13042	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ ( A ^ 1 ) )
46	5	recnd 10068	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
47	46	exp1d 13003	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 1 ) = A )
48	45, 47	breqtrd 4679	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ A )
49	13, 8, 5, 31, 48	ltletrd 10197	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < A )
50	13, 5	posdifd 10614	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( ( A ^ 3 ) / 3 ) < A <-> 0 < ( A - ( ( A ^ 3 ) / 3 ) ) ) )
51	49, 50	mpbid 222	. 2 |- ( A e. ( 0 (,] 1 ) -> 0 < ( A - ( ( A ^ 3 ) / 3 ) ) )
52		sin01bnd 14915	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) /\ ( sin ` A ) < A ) )
53	52	simpld 475	. 2 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) )
54	5, 13	resubcld 10458	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) e. RR )
55	5	resincld 14873	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( sin ` A ) e. RR )
56		lttr 10114	. . . 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 ) ) )
57	40, 56	mp3an1 1411	. . 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 ) ) )
58	54, 55, 57	syl2anc 693	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( 0 < ( A - ( ( A ^ 3 ) / 3 ) ) /\ ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) ) -> 0 < ( sin ` A ) ) )
59	51, 53, 58	mp2and 715	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   3c3 11071   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   (,]cioc 12176   ^cexp 12860   sincsin 14794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioc 12180  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800

This theorem is referenced by:  sin02gt0  14922  sincos1sgn  14923  sincos4thpi  24265