sincosq3sgn - Metamath Proof Explorer

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Theorem sincosq3sgn 26829

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

**Assertion**
Ref	Expression
sincosq3sgn	⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))

**Proof of Theorem sincosq3sgn**
Step	Hyp	Ref	Expression
1		pire 26783	. . 3 ⊢ π ∈ ℝ
2		3re 12423	. . . 4 ⊢ 3 ∈ ℝ
3		halfpire 26793	. . . 4 ⊢ (π / 2) ∈ ℝ
4	2, 3	remulcli 11325	. . 3 ⊢ (3 · (π / 2)) ∈ ℝ
5		rexr 11355	. . . 4 ⊢ (π ∈ ℝ → π ∈ ℝ^*)
6		rexr 11355	. . . 4 ⊢ ((3 · (π / 2)) ∈ ℝ → (3 · (π / 2)) ∈ ℝ^*)
7		elioo2 13517	. . . 4 ⊢ ((π ∈ ℝ^* ∧ (3 · (π / 2)) ∈ ℝ^*) → (𝐴 ∈ (π(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2)))))
8	5, 6, 7	syl2an 608	. . 3 ⊢ ((π ∈ ℝ ∧ (3 · (π / 2)) ∈ ℝ) → (𝐴 ∈ (π(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2)))))
9	1, 4, 8	mp2an 705	. 2 ⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
10		pidiv2halves 26796	. . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π
11	10	breq1i 5109	. . . . . . . 8 ⊢ (((π / 2) + (π / 2)) < 𝐴 ↔ π < 𝐴)
12		ltaddsub 11790	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((π / 2) + (π / 2)) < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
13	3, 3, 12	mp3an12 1480	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (((π / 2) + (π / 2)) < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
14	11, 13	bitr3id 288	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
15		ltsubadd 11786	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → ((𝐴 − (π / 2)) < π ↔ 𝐴 < (π + (π / 2))))
16	3, 1, 15	mp3an23 1482	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (π / 2)) < π ↔ 𝐴 < (π + (π / 2))))
17		df-3 12406	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
18	17	oveq1i 7430	. . . . . . . . . 10 ⊢ (3 · (π / 2)) = ((2 + 1) · (π / 2))
19		2cn 12418	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		ax-1cn 11258	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
21	3	recni 11323	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℂ
22	19, 20, 21	adddiri 11322	. . . . . . . . . 10 ⊢ ((2 + 1) · (π / 2)) = ((2 · (π / 2)) + (1 · (π / 2)))
23	1	recni 11323	. . . . . . . . . . . 12 ⊢ π ∈ ℂ
24		2ne0 12449	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
25	23, 19, 24	divcan2i 12060	. . . . . . . . . . 11 ⊢ (2 · (π / 2)) = π
26	21	mullidi 11314	. . . . . . . . . . 11 ⊢ (1 · (π / 2)) = (π / 2)
27	25, 26	oveq12i 7432	. . . . . . . . . 10 ⊢ ((2 · (π / 2)) + (1 · (π / 2))) = (π + (π / 2))
28	18, 22, 27	3eqtrri 2788	. . . . . . . . 9 ⊢ (π + (π / 2)) = (3 · (π / 2))
29	28	breq2i 5110	. . . . . . . 8 ⊢ (𝐴 < (π + (π / 2)) ↔ 𝐴 < (3 · (π / 2)))
30	16, 29	bitr2di 291	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 < (3 · (π / 2)) ↔ (𝐴 − (π / 2)) < π))
31	14, 30	anbi12d 644	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) ↔ ((π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π)))
32		resubcl 11622	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 − (π / 2)) ∈ ℝ)
33	3, 32	mpan2 704	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (π / 2)) ∈ ℝ)
34		sincosq2sgn 26828	. . . . . . . . 9 ⊢ ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) → (0 < (sin‘(𝐴 − (π / 2))) ∧ (cos‘(𝐴 − (π / 2))) < 0))
35		rexr 11355	. . . . . . . . . . 11 ⊢ ((π / 2) ∈ ℝ → (π / 2) ∈ ℝ^*)
36		elioo2 13517	. . . . . . . . . . 11 ⊢ (((π / 2) ∈ ℝ^* ∧ π ∈ ℝ^*) → ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) ↔ ((𝐴 − (π / 2)) ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π)))
37	35, 5, 36	syl2an 608	. . . . . . . . . 10 ⊢ (((π / 2) ∈ ℝ ∧ π ∈ ℝ) → ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) ↔ ((𝐴 − (π / 2)) ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π)))
38	3, 1, 37	mp2an 705	. . . . . . . . 9 ⊢ ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) ↔ ((𝐴 − (π / 2)) ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π))
39		ancom 466	. . . . . . . . 9 ⊢ ((0 < (sin‘(𝐴 − (π / 2))) ∧ (cos‘(𝐴 − (π / 2))) < 0) ↔ ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2)))))
40	34, 38, 39	3imtr3i 294	. . . . . . . 8 ⊢ (((𝐴 − (π / 2)) ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2)))))
41	33, 40	syl3an1 1181	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2)))))
42	41	3expib 1140	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (((π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2))))))
43	31, 42	sylbid 243	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2))))))
44	33	resincld 16311	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘(𝐴 − (π / 2))) ∈ ℝ)
45	44	lt0neg2d 11886	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 < (sin‘(𝐴 − (π / 2))) ↔ -(sin‘(𝐴 − (π / 2))) < 0))
46	45	anbi2d 642	. . . . 5 ⊢ (𝐴 ∈ ℝ → (((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2)))) ↔ ((cos‘(𝐴 − (π / 2))) < 0 ∧ -(sin‘(𝐴 − (π / 2))) < 0)))
47	43, 46	sylibd 242	. . . 4 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ -(sin‘(𝐴 − (π / 2))) < 0)))
48		recn 11290	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
49		pncan3 11565	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) + (𝐴 − (π / 2))) = 𝐴)
50	21, 48, 49	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((π / 2) + (𝐴 − (π / 2))) = 𝐴)
51	50	fveq2d 6889	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (sin‘𝐴))
52	33	recnd 11337	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (π / 2)) ∈ ℂ)
53		sinhalfpip 26821	. . . . . . . 8 ⊢ ((𝐴 − (π / 2)) ∈ ℂ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘(𝐴 − (π / 2))))
54	52, 53	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘(𝐴 − (π / 2))))
55	51, 54	eqtr3d 2797	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (cos‘(𝐴 − (π / 2))))
56	55	breq1d 5112	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴) < 0 ↔ (cos‘(𝐴 − (π / 2))) < 0))
57	50	fveq2d 6889	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘𝐴))
58		coshalfpip 26823	. . . . . . . 8 ⊢ ((𝐴 − (π / 2)) ∈ ℂ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = -(sin‘(𝐴 − (π / 2))))
59	52, 58	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = -(sin‘(𝐴 − (π / 2))))
60	57, 59	eqtr3d 2797	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = -(sin‘(𝐴 − (π / 2))))
61	60	breq1d 5112	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴) < 0 ↔ -(sin‘(𝐴 − (π / 2))) < 0))
62	56, 61	anbi12d 644	. . . 4 ⊢ (𝐴 ∈ ℝ → (((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0) ↔ ((cos‘(𝐴 − (π / 2))) < 0 ∧ -(sin‘(𝐴 − (π / 2))) < 0)))
63	47, 62	sylibrd 262	. . 3 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0)))
64	63	3impib 1134	. 2 ⊢ ((𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))
65	9, 64	sylbi 220	1 ⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 ℝ^*cxr 11342 < clt 11343 − cmin 11541 -cneg 11542 / cdiv 11973 2c2 12397 3c3 12398 (,)cioo 13476 sincsin 16229 cosccos 16230 πcpi 16232

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-fac 14418 df-bc 14447 df-hash 14475 df-shft 15220 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-limsup 15638 df-clim 15655 df-rlim 15656 df-sum 15854 df-ef 16233 df-sin 16235 df-cos 16236 df-pi 16238 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-mulg 19278 df-cntz 19531 df-cmn 19996 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-limc 26186 df-dv 26187

This theorem is used by: sincosq4sgn 26830

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