Theorem sincosq1sgn 8640

Description: The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

Assertion

Ref	Expression
sincosq1sgn	|- (A e. (0(,)(pi / 2)) -> (0 < (sin` A) /\ 0 < (cos` A)))

Proof of Theorem sincosq1sgn

Step	Hyp	Ref	Expression
1		0re 5420	. . 3 |- 0 e. RR
2		pire 8615	. . . 4 |- pi e. RR
3		2re 5934	. . . 4 |- 2 e. RR
4		2ne0 5945	. . . 4 |- 2 =/= 0
5	2, 3, 4	redivcl 5762	. . 3 |- (pi / 2) e. RR
6		elioo2t 6324	. . . 4 |- ((0 e. RR* /\ (pi / 2) e. RR*) -> (A e. (0(,)(pi / 2)) <-> (A e. RR /\ 0 < A /\ A < (pi / 2))))
7		rexrt 5479	. . . 4 |- (0 e. RR -> 0 e. RR*)
8		rexrt 5479	. . . 4 |- ((pi / 2) e. RR -> (pi / 2) e. RR*)
9	6, 7, 8	syl2an 454	. . 3 |- ((0 e. RR /\ (pi / 2) e. RR) -> (A e. (0(,)(pi / 2)) <-> (A e. RR /\ 0 < A /\ A < (pi / 2))))
10	1, 5, 9	mp2an 696	. 2 |- (A e. (0(,)(pi / 2)) <-> (A e. RR /\ 0 < A /\ A < (pi / 2)))
11		sincosq1lem 8639	. . 3 |- ((A e. RR /\ 0 < A /\ A < (pi / 2)) -> 0 < (sin` A))
12		sincosq1lem 8639	. . . . . . 7 |- ((((pi / 2) - A) e. RR /\ 0 < ((pi / 2) - A) /\ ((pi / 2) - A) < (pi / 2)) -> 0 < (sin` ((pi / 2) - A)))
13		resubclt 5418	. . . . . . . 8 |- (((pi / 2) e. RR /\ A e. RR) -> ((pi / 2) - A) e. RR)
14	5, 13	mpan 694	. . . . . . 7 |- (A e. RR -> ((pi / 2) - A) e. RR)
15	12, 14	syl3an1 858	. . . . . 6 |- ((A e. RR /\ 0 < ((pi / 2) - A) /\ ((pi / 2) - A) < (pi / 2)) -> 0 < (sin` ((pi / 2) - A)))
16	15	3expib 835	. . . . 5 |- (A e. RR -> ((0 < ((pi / 2) - A) /\ ((pi / 2) - A) < (pi / 2)) -> 0 < (sin` ((pi / 2) - A))))
17		ltsub13t 5624	. . . . . . . . 9 |- ((0 e. RR /\ (pi / 2) e. RR /\ A e. RR) -> (0 < ((pi / 2) - A) <-> A < ((pi / 2) - 0)))
18	1, 5, 17	mp3an12 904	. . . . . . . 8 |- (A e. RR -> (0 < ((pi / 2) - A) <-> A < ((pi / 2) - 0)))
19	5	recn 5294	. . . . . . . . . 10 |- (pi / 2) e. CC
20	19	subid1 5372	. . . . . . . . 9 |- ((pi / 2) - 0) = (pi / 2)
21	20	breq2i 2622	. . . . . . . 8 |- (A < ((pi / 2) - 0) <-> A < (pi / 2))
22	18, 21	syl6bb 535	. . . . . . 7 |- (A e. RR -> (0 < ((pi / 2) - A) <-> A < (pi / 2)))
23		ltsub23t 5623	. . . . . . . . 9 |- (((pi / 2) e. RR /\ A e. RR /\ (pi / 2) e. RR) -> (((pi / 2) - A) < (pi / 2) <-> ((pi / 2) - (pi / 2)) < A))
24	5, 5, 23	mp3an13 905	. . . . . . . 8 |- (A e. RR -> (((pi / 2) - A) < (pi / 2) <-> ((pi / 2) - (pi / 2)) < A))
25	19	subid 5371	. . . . . . . . 9 |- ((pi / 2) - (pi / 2)) = 0
26	25	breq1i 2621	. . . . . . . 8 |- (((pi / 2) - (pi / 2)) < A <-> 0 < A)
27	24, 26	syl6bb 535	. . . . . . 7 |- (A e. RR -> (((pi / 2) - A) < (pi / 2) <-> 0 < A))
28	22, 27	anbi12d 627	. . . . . 6 |- (A e. RR -> ((0 < ((pi / 2) - A) /\ ((pi / 2) - A) < (pi / 2)) <-> (A < (pi / 2) /\ 0 < A)))
29		ancom 435	. . . . . 6 |- ((A < (pi / 2) /\ 0 < A) <-> (0 < A /\ A < (pi / 2)))
30	28, 29	syl6bb 535	. . . . 5 |- (A e. RR -> ((0 < ((pi / 2) - A) /\ ((pi / 2) - A) < (pi / 2)) <-> (0 < A /\ A < (pi / 2))))
31		recnt 5293	. . . . . . 7 |- (A e. RR -> A e. CC)
32		sinhalfpim 8636	. . . . . . 7 |- (A e. CC -> (sin` ((pi / 2) - A)) = (cos` A))
33	31, 32	syl 10	. . . . . 6 |- (A e. RR -> (sin` ((pi / 2) - A)) = (cos` A))
34	33	breq2d 2625	. . . . 5 |- (A e. RR -> (0 < (sin` ((pi / 2) - A)) <-> 0 < (cos` A)))
35	16, 30, 34	3imtr3d 541	. . . 4 |- (A e. RR -> ((0 < A /\ A < (pi / 2)) -> 0 < (cos` A)))
36	35	3impib 830	. . 3 |- ((A e. RR /\ 0 < A /\ A < (pi / 2)) -> 0 < (cos` A))
37	11, 36	jca 288	. 2 |- ((A e. RR /\ 0 < A /\ A < (pi / 2)) -> (0 < (sin` A) /\ 0 < (cos` A)))
38	10, 37	sylbi 199	1 |- (A e. (0(,)(pi / 2)) -> (0 < (sin` A) /\ 0 < (cos` A)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   class class class wbr 2614  ` cfv 3177  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214   - cmin 5272   / cdiv 5274  RR*cxr 5465   < clt 5466  2c2 5916  (,)cioo 6302  sincsin 7245  cosccos 7246  picpi 7247

This theorem is referenced by:  sincosq2sgn 8641  sincos6thpi 8647

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-iin 2564  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-map 4314  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-r1 4623  df-rank 4624  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-3 5926  df-4 5927  df-5 5928  df-6 5929  df-7 5930  df-8 5931  df-9 5932  df-n0 6055  df-z 6091  df-fl 6180  df-q 6202  df-rp 6227  df-seq1 6253  df-shft 6286  df-ioo 6306  df-ioc 6307  df-ico 6308  df-icc 6309  df-uz 6358  df-fz 6408  df-seqz 6473  df-seq0 6474  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-fac 6877  df-bc 6902  df-clim 6921  df-sum 6926  df-cncf 7206  df-ef 7248  df-sin 7250  df-cos 7251  df-pi 7252  df-top 7542  df-cn 7704  df-cnp 7705  df-met 7743  df-bl 7745  df-opn 7746  df-lm 7874

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