Theorem sin02gt0 8744

Description: The sine of a positive real number less than or equal to 2 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)

Assertion

Ref	Expression
sin02gt0	|- (A e. (0(,]2) -> 0 < (sin` A))

Proof of Theorem sin02gt0

Step	Hyp	Ref	Expression
1		rehalfcl 7220	. . . . . . . . 9 |- (A e. RR -> (A / 2) e. RR)
2	1	3ad2ant1 897	. . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> (A / 2) e. RR)
3		2re 7163	. . . . . . . . . 10 |- 2 e. RR
4		2pos 7173	. . . . . . . . . 10 |- 0 < 2
5		divgt0 7037	. . . . . . . . . 10 |- (((A e. RR /\ 0 < A) /\ (2 e. RR /\ 0 < 2)) -> 0 < (A / 2))
6	3, 4, 5	mpanr12 778	. . . . . . . . 9 |- ((A e. RR /\ 0 < A) -> 0 < (A / 2))
7	6	3adant3 896	. . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> 0 < (A / 2))
8		lediv1OLD 7034	. . . . . . . . . . . . 13 |- (((A e. RR /\ 2 e. RR /\ 2 e. RR) /\ 0 < 2) -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
9	4, 8	mpan2 760	. . . . . . . . . . . 12 |- ((A e. RR /\ 2 e. RR /\ 2 e. RR) -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
10	3, 3, 9	mp3an23 1183	. . . . . . . . . . 11 |- (A e. RR -> (A <_ 2 <-> (A / 2) <_ (2 / 2)))
11	10	biimpa 460	. . . . . . . . . 10 |- ((A e. RR /\ A <_ 2) -> (A / 2) <_ (2 / 2))
12		2cn 7164	. . . . . . . . . . 11 |- 2 e. CC
13		2ne0 7174	. . . . . . . . . . 11 |- 2 =/= 0
14	12, 13	dividi 6946	. . . . . . . . . 10 |- (2 / 2) = 1
15	11, 14	syl6breq 3376	. . . . . . . . 9 |- ((A e. RR /\ A <_ 2) -> (A / 2) <_ 1)
16	15	3adant2 895	. . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> (A / 2) <_ 1)
17	2, 7, 16	3jca 1050	. . . . . . 7 |- ((A e. RR /\ 0 < A /\ A <_ 2) -> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1))
18		0re 6603	. . . . . . . 8 |- 0 e. RR
19		elioc2 7558	. . . . . . . 8 |- ((0 e. RR /\ 2 e. RR) -> (A e. (0(,]2) <-> (A e. RR /\ 0 < A /\ A <_ 2)))
20	18, 3, 19	mp2an 761	. . . . . . 7 |- (A e. (0(,]2) <-> (A e. RR /\ 0 < A /\ A <_ 2))
21		1re 6598	. . . . . . . 8 |- 1 e. RR
22		elioc2 7558	. . . . . . . 8 |- ((0 e. RR /\ 1 e. RR) -> ((A / 2) e. (0(,]1) <-> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1)))
23	18, 21, 22	mp2an 761	. . . . . . 7 |- ((A / 2) e. (0(,]1) <-> ((A / 2) e. RR /\ 0 < (A / 2) /\ (A / 2) <_ 1))
24	17, 20, 23	3imtr4i 236	. . . . . 6 |- (A e. (0(,]2) -> (A / 2) e. (0(,]1))
25		sin01gt0 8742	. . . . . 6 |- ((A / 2) e. (0(,]1) -> 0 < (sin` (A / 2)))
26	24, 25	syl 12	. . . . 5 |- (A e. (0(,]2) -> 0 < (sin` (A / 2)))
27		cos01gt0 8743	. . . . . 6 |- ((A / 2) e. (0(,]1) -> 0 < (cos` (A / 2)))
28	24, 27	syl 12	. . . . 5 |- (A e. (0(,]2) -> 0 < (cos` (A / 2)))
29	20, 2	sylbi 216	. . . . . 6 |- (A e. (0(,]2) -> (A / 2) e. RR)
30		resincl 8703	. . . . . . 7 |- ((A / 2) e. RR -> (sin` (A / 2)) e. RR)
31		recoscl 8704	. . . . . . 7 |- ((A / 2) e. RR -> (cos` (A / 2)) e. RR)
32		axmulgt0 6675	. . . . . . 7 |- (((sin` (A / 2)) e. RR /\ (cos` (A / 2)) e. RR) -> ((0 < (sin` (A / 2)) /\ 0 < (cos` (A / 2))) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2)))))
33	30, 31, 32	syl11anc 524	. . . . . 6 |- ((A / 2) e. RR -> ((0 < (sin` (A / 2)) /\ 0 < (cos` (A / 2))) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2)))))
34	29, 33	syl 12	. . . . 5 |- (A e. (0(,]2) -> ((0 < (sin` (A / 2)) /\ 0 < (cos` (A / 2))) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2)))))
35	26, 28, 34	mp2and 767	. . . 4 |- (A e. (0(,]2) -> 0 < ((sin` (A / 2)) x. (cos` (A / 2))))
36		remulcl 6457	. . . . . 6 |- (((sin` (A / 2)) e. RR /\ (cos` (A / 2)) e. RR) -> ((sin` (A / 2)) x. (cos` (A / 2))) e. RR)
37	30, 31, 36	syl11anc 524	. . . . 5 |- ((A / 2) e. RR -> ((sin` (A / 2)) x. (cos` (A / 2))) e. RR)
38		axmulgt0 6675	. . . . . . 7 |- ((2 e. RR /\ ((sin` (A / 2)) x. (cos` (A / 2))) e. RR) -> ((0 < 2 /\ 0 < ((sin` (A / 2)) x. (cos` (A / 2)))) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2))))))
39	3, 38	mpan 759	. . . . . 6 |- (((sin` (A / 2)) x. (cos` (A / 2))) e. RR -> ((0 < 2 /\ 0 < ((sin` (A / 2)) x. (cos` (A / 2)))) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2))))))
40	4, 39	mpani 762	. . . . 5 |- (((sin` (A / 2)) x. (cos` (A / 2))) e. RR -> (0 < ((sin` (A / 2)) x. (cos` (A / 2))) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2))))))
41	29, 37, 40	3syl 24	. . . 4 |- (A e. (0(,]2) -> (0 < ((sin` (A / 2)) x. (cos` (A / 2))) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2))))))
42	35, 41	mpd 29	. . 3 |- (A e. (0(,]2) -> 0 < (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
43	29	recnd 6468	. . . 4 |- (A e. (0(,]2) -> (A / 2) e. CC)
44		sin2t 8727	. . . 4 |- ((A / 2) e. CC -> (sin` (2 x. (A / 2))) = (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
45	43, 44	syl 12	. . 3 |- (A e. (0(,]2) -> (sin` (2 x. (A / 2))) = (2 x. ((sin` (A / 2)) x. (cos` (A / 2)))))
46	42, 45	breqtrrd 3363	. 2 |- (A e. (0(,]2) -> 0 < (sin` (2 x. (A / 2))))
47	20	simp1bi 891	. . . . 5 |- (A e. (0(,]2) -> A e. RR)
48	47	recnd 6468	. . . 4 |- (A e. (0(,]2) -> A e. CC)
49		divcan2 6910	. . . . 5 |- ((A e. CC /\ 2 e. CC /\ 2 =/= 0) -> (2 x. (A / 2)) = A)
50	12, 13, 49	mp3an23 1183	. . . 4 |- (A e. CC -> (2 x. (A / 2)) = A)
51	48, 50	syl 12	. . 3 |- (A e. (0(,]2) -> (2 x. (A / 2)) = A)
52	51	fveq2d 4685	. 2 |- (A e. (0(,]2) -> (sin` (2 x. (A / 2))) = (sin` A))
53	46, 52	breqtrd 3361	1 |- (A e. (0(,]2) -> 0 < (sin` A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447   <_ cle 6448   < clt 6653  2c2 7145  (,]cioc 7525  sincsin 8557  cosccos 8558

This theorem is referenced by:  sincos2sgn 8746  pilem1 10020  sinhalfpilem 10028  sincosq1lem 10052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-3 7155  df-4 7156  df-5 7157  df-6 7158  df-7 7159  df-8 7160  df-n0 7309  df-z 7345  df-fl 7463  df-ioc 7529  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-fac 8184  df-bc 8209  df-clim 8235  df-sum 8240  df-ef 8560  df-sin 8562  df-cos 8563

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