Theorem sin01gt0 12403

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 8285	. . . . . . . 8 |- 0 e. RR*
2		1re 8238	. . . . . . . 8 |- 1 e. RR
3		elioc2 10232	. . . . . . . 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 1039	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6		3nn0 9479	. . . . . 6 |- 3 e. NN0
7		reexpcl 10881	. . . . . 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 9276	. . . . . 6 |- 3 e. RR
10		3ap0 9298	. . . . . 6 |- 3 # 0
11		redivclap 8970	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
12	9, 10, 11	mp3an23 1366	. . . . 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 9569	. . . . . . . . 9 |- 3 e. ZZ
15		expgt0 10897	. . . . . . . . 9 |- ( ( A e. RR /\ 3 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 3 ) )
16	14, 15	mp3an2 1362	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 3 ) )
17	16	3adant3 1044	. . . . . . 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 8365	. . . . . . . 8 |- 0 < 1
20	2, 19	pm3.2i 272	. . . . . . 7 |- ( 1 e. RR /\ 0 < 1 )
21		3pos 9296	. . . . . . . 8 |- 0 < 3
22	9, 21	pm3.2i 272	. . . . . . 7 |- ( 3 e. RR /\ 0 < 3 )
23		1lt3 9374	. . . . . . . 8 |- 1 < 3
24		ltdiv2 9126	. . . . . . . 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 1364	. . . . . 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 8267	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. CC )
29	28	div1d 9019	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 1 ) = ( A ^ 3 ) )
30	27, 29	breqtrd 4119	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( A ^ 3 ) )
31		1nn0 9477	. . . . . . 7 |- 1 e. NN0
32	31	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 1 e. NN0 )
33		1le3 9414	. . . . . . . 8 |- 1 <_ 3
34		1z 9566	. . . . . . . . 9 |- 1 e. ZZ
35	34	eluz1i 9824	. . . . . . . 8 |- ( 3 e. ( ZZ>= ` 1 ) <-> ( 3 e. ZZ /\ 1 <_ 3 ) )
36	14, 33, 35	mpbir2an 951	. . . . . . 7 |- 3 e. ( ZZ>= ` 1 )
37	36	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 3 e. ( ZZ>= ` 1 ) )
38	4	simp2bi 1040	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
39		0re 8239	. . . . . . . 8 |- 0 e. RR
40		ltle 8326	. . . . . . . 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 1041	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
44	5, 32, 37, 42, 43	leexp2rd 11028	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ ( A ^ 1 ) )
45	5	recnd 8267	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
46	45	exp1d 10993	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 1 ) = A )
47	44, 46	breqtrd 4119	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ A )
48	13, 8, 5, 30, 47	ltletrd 8662	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < A )
49	13, 5	posdifd 8771	. . 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 12398	. . 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 8619	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) e. RR )
54	5	resincld 12364	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( sin ` A ) e. RR )
55		lttr 8312	. . 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 1379	. 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 1005   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   RR*cxr 8272   < clt 8273   <_ cle 8274   - cmin 8409   # cap 8820   / cdiv 8911   3c3 9254   NN0cn0 9461   ZZcz 9540   ZZ>=cuz 9816   (,]cioc 10185   ^cexp 10863   sincsin 12285

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ioc 10189  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-ihash 11101  df-shft 11455  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289  df-sin 12291

This theorem is referenced by:  sin02gt0  12405  sincos1sgn  12406  sincos4thpi  15651