Theorem sin01gt0 12507

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 8362	. . . . . . . 8 |- 0 e. RR*
2		1re 8315	. . . . . . . 8 |- 1 e. RR
3		elioc2 10317	. . . . . . . 8 |- ( ( 0 e. RR* /\ 1 e. RR ) -> ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) ) )
4	1, 2, 3	mp2an 430	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) <-> ( A e. RR /\ 0 < A /\ A <_ 1 ) )
5	4	simp1bi 1043	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. RR )
6		3nn0 9560	. . . . . 6 |- 3 e. NN0
7		reexpcl 10971	. . . . . 6 |- ( ( A e. RR /\ 3 e. NN0 ) -> ( A ^ 3 ) e. RR )
8	5, 6, 7	sylancl 417	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. RR )
9		3re 9357	. . . . . 6 |- 3 e. RR
10		3ap0 9379	. . . . . 6 |- 3 # 0
11		redivclap 9051	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 3 e. RR /\ 3 # 0 ) -> ( ( A ^ 3 ) / 3 ) e. RR )
12	9, 10, 11	mp3an23 1370	. . . . 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 9652	. . . . . . . . 9 |- 3 e. ZZ
15		expgt0 10987	. . . . . . . . 9 |- ( ( A e. RR /\ 3 e. ZZ /\ 0 < A ) -> 0 < ( A ^ 3 ) )
16	14, 15	mp3an2 1366	. . . . . . . 8 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( A ^ 3 ) )
17	16	3adant3 1048	. . . . . . 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 8443	. . . . . . . 8 |- 0 < 1
20	2, 19	pm3.2i 272	. . . . . . 7 |- ( 1 e. RR /\ 0 < 1 )
21		3pos 9377	. . . . . . . 8 |- 0 < 3
22	9, 21	pm3.2i 272	. . . . . . 7 |- ( 3 e. RR /\ 0 < 3 )
23		1lt3 9455	. . . . . . . 8 |- 1 < 3
24		ltdiv2 9207	. . . . . . . 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 1368	. . . . . 6 |- ( ( ( A ^ 3 ) e. RR /\ 0 < ( A ^ 3 ) ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
27	8, 18, 26	syl2anc 415	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( ( A ^ 3 ) / 1 ) )
28	8	recnd 8344	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) e. CC )
29	28	div1d 9100	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 1 ) = ( A ^ 3 ) )
30	27, 29	breqtrd 4151	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < ( A ^ 3 ) )
31		1nn0 9558	. . . . . . 7 |- 1 e. NN0
32	31	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 1 e. NN0 )
33		1le3 9495	. . . . . . . 8 |- 1 <_ 3
34		1z 9649	. . . . . . . . 9 |- 1 e. ZZ
35	34	eluz1i 9908	. . . . . . . 8 |- ( 3 e. ( ZZ>= ` 1 ) <-> ( 3 e. ZZ /\ 1 <_ 3 ) )
36	14, 33, 35	mpbir2an 955	. . . . . . 7 |- 3 e. ( ZZ>= ` 1 )
37	36	a1i 9	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 3 e. ( ZZ>= ` 1 ) )
38	4	simp2bi 1044	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> 0 < A )
39		0re 8316	. . . . . . . 8 |- 0 e. RR
40		ltle 8403	. . . . . . . 8 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A -> 0 <_ A ) )
41	39, 5, 40	sylancr 418	. . . . . . 7 |- ( A e. ( 0 (,] 1 ) -> ( 0 < A -> 0 <_ A ) )
42	38, 41	mpd 13	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> 0 <_ A )
43	4	simp3bi 1045	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A <_ 1 )
44	5, 32, 37, 42, 43	leexp2rd 11119	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ ( A ^ 1 ) )
45	5	recnd 8344	. . . . . 6 |- ( A e. ( 0 (,] 1 ) -> A e. CC )
46	45	exp1d 11084	. . . . 5 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 1 ) = A )
47	44, 46	breqtrd 4151	. . . 4 |- ( A e. ( 0 (,] 1 ) -> ( A ^ 3 ) <_ A )
48	13, 8, 5, 30, 47	ltletrd 8741	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( ( A ^ 3 ) / 3 ) < A )
49	13, 5	posdifd 8850	. . 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 12502	. . 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 8698	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( A - ( ( A ^ 3 ) / 3 ) ) e. RR )
54	5	resincld 12468	. . 3 |- ( A e. ( 0 (,] 1 ) -> ( sin ` A ) e. RR )
55		lttr 8389	. . 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 1383	. 2 |- ( A e. ( 0 (,] 1 ) -> ( ( 0 < ( A - ( ( A ^ 3 ) / 3 ) ) /\ ( A - ( ( A ^ 3 ) / 3 ) ) < ( sin ` A ) ) -> 0 < ( sin ` A ) ) )
57	50, 52, 56	mp2and 437	1 |- ( A e. ( 0 (,] 1 ) -> 0 < ( sin ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   RR*cxr 8349   < clt 8350   <_ cle 8351   - cmin 8487   # cap 8899   / cdiv 8992   3c3 9335   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   (,]cioc 10270   ^cexp 10953   sincsin 12389

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ioc 10274  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-sin 12395

This theorem is referenced by:  sin02gt0  12509  sincos1sgn  12510  sincos4thpi  15864