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Theorem sincosq1sgn 15549

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 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))

**Proof of Theorem sincosq1sgn**
Step	Hyp	Ref	Expression
1		0xr 8225	. . 3 ⊢ 0 ∈ ℝ^*
2		halfpire 15515	. . . 4 ⊢ (π / 2) ∈ ℝ
3	2	rexri 8236	. . 3 ⊢ (π / 2) ∈ ℝ^*
4		elioo2 10155	. . 3 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
5	1, 3, 4	mp2an 426	. 2 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
6		sincosq1lem 15548	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
7		resubcl 8442	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
8	2, 7	mpan 424	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((π / 2) − 𝐴) ∈ ℝ)
9		sincosq1lem 15548	. . . . . . 7 ⊢ ((((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
10	8, 9	syl3an1 1306	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
11	10	3expib 1232	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴))))
12		0re 8178	. . . . . . . . 9 ⊢ 0 ∈ ℝ
13		ltsub13 8622	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
14	12, 2, 13	mp3an12 1363	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
15	2	recni 8190	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
16	15	subid1i 8450	. . . . . . . . 9 ⊢ ((π / 2) − 0) = (π / 2)
17	16	breq2i 4096	. . . . . . . 8 ⊢ (𝐴 < ((π / 2) − 0) ↔ 𝐴 < (π / 2))
18	14, 17	bitrdi 196	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < (π / 2)))
19		ltsub23 8621	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
20	2, 2, 19	mp3an13 1364	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
21	15	subidi 8449	. . . . . . . . 9 ⊢ ((π / 2) − (π / 2)) = 0
22	21	breq1i 4095	. . . . . . . 8 ⊢ (((π / 2) − (π / 2)) < 𝐴 ↔ 0 < 𝐴)
23	20, 22	bitrdi 196	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < (π / 2) ↔ 0 < 𝐴))
24	18, 23	anbi12d 473	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) ↔ (𝐴 < (π / 2) ∧ 0 < 𝐴)))
25	24	biancomd 271	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) ↔ (0 < 𝐴 ∧ 𝐴 < (π / 2))))
26		recn 8164	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
27		sinhalfpim 15544	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
28	26, 27	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
29	28	breq2d 4100	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < (sin‘((π / 2) − 𝐴)) ↔ 0 < (cos‘𝐴)))
30	11, 25, 29	3imtr3d 202	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (cos‘𝐴)))
31	30	3impib 1227	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (cos‘𝐴))
32	6, 31	jca 306	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
33	5, 32	sylbi 121	1 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 ‘cfv 5326 (class class class)co 6017 ℂcc 8029 ℝcr 8030 0cc0 8031 ℝ^*cxr 8212 < clt 8213 − cmin 8349 / cdiv 8851 2c2 9193 (,)cioo 10122 sincsin 12204 cosccos 12205 πcpi 12207

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151 ax-pre-suploc 8152 ax-addf 8153 ax-mulf 8154

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-of 6234 df-1st 6302 df-2nd 6303 df-recs 6470 df-irdg 6535 df-frec 6556 df-1o 6581 df-oadd 6585 df-er 6701 df-map 6818 df-pm 6819 df-en 6909 df-dom 6910 df-fin 6911 df-sup 7182 df-inf 7183 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-9 9208 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-xneg 10006 df-xadd 10007 df-ioo 10126 df-ioc 10127 df-ico 10128 df-icc 10129 df-fz 10243 df-fzo 10377 df-seqfrec 10709 df-exp 10800 df-fac 10987 df-bc 11009 df-ihash 11037 df-shft 11375 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-clim 11839 df-sumdc 11914 df-ef 12208 df-sin 12210 df-cos 12211 df-pi 12213 df-rest 13323 df-topgen 13342 df-psmet 14556 df-xmet 14557 df-met 14558 df-bl 14559 df-mopn 14560 df-top 14721 df-topon 14734 df-bases 14766 df-ntr 14819 df-cn 14911 df-cnp 14912 df-tx 14976 df-cncf 15294 df-limced 15379 df-dvap 15380

This theorem is referenced by: sincosq2sgn 15550 tanrpcl 15560 tangtx 15561 sincos6thpi 15565

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