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Theorem sincosq1sgn 15620

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 8268	. . 3 ⊢ 0 ∈ ℝ^*
2		halfpire 15586	. . . 4 ⊢ (π / 2) ∈ ℝ
3	2	rexri 8279	. . 3 ⊢ (π / 2) ∈ ℝ^*
4		elioo2 10200	. . 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 15619	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
7		resubcl 8485	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
8	2, 7	mpan 424	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((π / 2) − 𝐴) ∈ ℝ)
9		sincosq1lem 15619	. . . . . . 7 ⊢ ((((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
10	8, 9	syl3an1 1307	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
11	10	3expib 1233	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴))))
12		0re 8222	. . . . . . . . 9 ⊢ 0 ∈ ℝ
13		ltsub13 8665	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
14	12, 2, 13	mp3an12 1364	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
15	2	recni 8234	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
16	15	subid1i 8493	. . . . . . . . 9 ⊢ ((π / 2) − 0) = (π / 2)
17	16	breq2i 4101	. . . . . . . 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 8664	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
20	2, 2, 19	mp3an13 1365	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
21	15	subidi 8492	. . . . . . . . 9 ⊢ ((π / 2) − (π / 2)) = 0
22	21	breq1i 4100	. . . . . . . 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 8208	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
27		sinhalfpim 15615	. . . . . . 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 4105	. . . . 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 1228	. . 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 1005 = wceq 1398 ∈ wcel 2202 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 ℝcr 8074 0cc0 8075 ℝ^*cxr 8255 < clt 8256 − cmin 8392 / cdiv 8894 2c2 9236 (,)cioo 10167 sincsin 12268 cosccos 12269 πcpi 12271

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 ax-pre-suploc 8196 ax-addf 8197 ax-mulf 8198

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-disj 4070 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-oadd 6629 df-er 6745 df-map 6862 df-pm 6863 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248 df-7 9249 df-8 9250 df-9 9251 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-xneg 10051 df-xadd 10052 df-ioo 10171 df-ioc 10172 df-ico 10173 df-icc 10174 df-fz 10289 df-fzo 10423 df-seqfrec 10756 df-exp 10847 df-fac 11034 df-bc 11056 df-ihash 11084 df-shft 11438 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-clim 11902 df-sumdc 11977 df-ef 12272 df-sin 12274 df-cos 12275 df-pi 12277 df-rest 13387 df-topgen 13406 df-psmet 14622 df-xmet 14623 df-met 14624 df-bl 14625 df-mopn 14626 df-top 14792 df-topon 14805 df-bases 14837 df-ntr 14890 df-cn 14982 df-cnp 14983 df-tx 15047 df-cncf 15365 df-limced 15450 df-dvap 15451

This theorem is referenced by: sincosq2sgn 15621 tanrpcl 15631 tangtx 15632 sincos6thpi 15636

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