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Theorem sincosq1sgn 15569

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 8226	. . 3 ⊢ 0 ∈ ℝ^*
2		halfpire 15535	. . . 4 ⊢ (π / 2) ∈ ℝ
3	2	rexri 8237	. . 3 ⊢ (π / 2) ∈ ℝ^*
4		elioo2 10156	. . 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 15568	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
7		resubcl 8443	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
8	2, 7	mpan 424	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((π / 2) − 𝐴) ∈ ℝ)
9		sincosq1lem 15568	. . . . . . 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 8179	. . . . . . . . 9 ⊢ 0 ∈ ℝ
13		ltsub13 8623	. . . . . . . . 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 8191	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
16	15	subid1i 8451	. . . . . . . . 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 8622	. . . . . . . . 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 8450	. . . . . . . . 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 8165	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
27		sinhalfpim 15564	. . . . . . 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 6018 ℂcc 8030 ℝcr 8031 0cc0 8032 ℝ^*cxr 8213 < clt 8214 − cmin 8350 / cdiv 8852 2c2 9194 (,)cioo 10123 sincsin 12223 cosccos 12224 πcpi 12226

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 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

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 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ioc 10128 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11393 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-clim 11857 df-sumdc 11932 df-ef 12227 df-sin 12229 df-cos 12230 df-pi 12232 df-rest 13342 df-topgen 13361 df-psmet 14576 df-xmet 14577 df-met 14578 df-bl 14579 df-mopn 14580 df-top 14741 df-topon 14754 df-bases 14786 df-ntr 14839 df-cn 14931 df-cnp 14932 df-tx 14996 df-cncf 15314 df-limced 15399 df-dvap 15400

This theorem is referenced by: sincosq2sgn 15570 tanrpcl 15580 tangtx 15581 sincos6thpi 15585

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