sincosq3sgn - Metamath Proof Explorer

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Theorem sincosq3sgn 26001

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 25959	. . 3 ⊢ π ∈ ℝ
2		3re 12288	. . . 4 ⊢ 3 ∈ ℝ
3		halfpire 25965	. . . 4 ⊢ (π / 2) ∈ ℝ
4	2, 3	remulcli 11226	. . 3 ⊢ (3 · (π / 2)) ∈ ℝ
5		rexr 11256	. . . 4 ⊢ (π ∈ ℝ → π ∈ ℝ^*)
6		rexr 11256	. . . 4 ⊢ ((3 · (π / 2)) ∈ ℝ → (3 · (π / 2)) ∈ ℝ^*)
7		elioo2 13361	. . . 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 596	. . 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 690	. 2 ⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
10		pidiv2halves 25968	. . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π
11	10	breq1i 5154	. . . . . . . 8 ⊢ (((π / 2) + (π / 2)) < 𝐴 ↔ π < 𝐴)
12		ltaddsub 11684	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((π / 2) + (π / 2)) < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
13	3, 3, 12	mp3an12 1451	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (((π / 2) + (π / 2)) < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
14	11, 13	bitr3id 284	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (π < 𝐴 ↔ (π / 2) < (𝐴 − (π / 2))))
15		ltsubadd 11680	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → ((𝐴 − (π / 2)) < π ↔ 𝐴 < (π + (π / 2))))
16	3, 1, 15	mp3an23 1453	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((𝐴 − (π / 2)) < π ↔ 𝐴 < (π + (π / 2))))
17		df-3 12272	. . . . . . . . . . 11 ⊢ 3 = (2 + 1)
18	17	oveq1i 7415	. . . . . . . . . 10 ⊢ (3 · (π / 2)) = ((2 + 1) · (π / 2))
19		2cn 12283	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
20		ax-1cn 11164	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
21	3	recni 11224	. . . . . . . . . . 11 ⊢ (π / 2) ∈ ℂ
22	19, 20, 21	adddiri 11223	. . . . . . . . . 10 ⊢ ((2 + 1) · (π / 2)) = ((2 · (π / 2)) + (1 · (π / 2)))
23	1	recni 11224	. . . . . . . . . . . 12 ⊢ π ∈ ℂ
24		2ne0 12312	. . . . . . . . . . . 12 ⊢ 2 ≠ 0
25	23, 19, 24	divcan2i 11953	. . . . . . . . . . 11 ⊢ (2 · (π / 2)) = π
26	21	mullidi 11215	. . . . . . . . . . 11 ⊢ (1 · (π / 2)) = (π / 2)
27	25, 26	oveq12i 7417	. . . . . . . . . 10 ⊢ ((2 · (π / 2)) + (1 · (π / 2))) = (π + (π / 2))
28	18, 22, 27	3eqtrri 2765	. . . . . . . . 9 ⊢ (π + (π / 2)) = (3 · (π / 2))
29	28	breq2i 5155	. . . . . . . 8 ⊢ (𝐴 < (π + (π / 2)) ↔ 𝐴 < (3 · (π / 2)))
30	16, 29	bitr2di 287	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 < (3 · (π / 2)) ↔ (𝐴 − (π / 2)) < π))
31	14, 30	anbi12d 631	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) ↔ ((π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π)))
32		resubcl 11520	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 − (π / 2)) ∈ ℝ)
33	3, 32	mpan2 689	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (π / 2)) ∈ ℝ)
34		sincosq2sgn 26000	. . . . . . . . 9 ⊢ ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) → (0 < (sin‘(𝐴 − (π / 2))) ∧ (cos‘(𝐴 − (π / 2))) < 0))
35		rexr 11256	. . . . . . . . . . 11 ⊢ ((π / 2) ∈ ℝ → (π / 2) ∈ ℝ^*)
36		elioo2 13361	. . . . . . . . . . 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 596	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ ((𝐴 − (π / 2)) ∈ ((π / 2)(,)π) ↔ ((𝐴 − (π / 2)) ∈ ℝ ∧ (π / 2) < (𝐴 − (π / 2)) ∧ (𝐴 − (π / 2)) < π))
39		ancom 461	. . . . . . . . 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 290	. . . . . . . 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 1163	. . . . . . 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 1122	. . . . . 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 239	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ 0 < (sin‘(𝐴 − (π / 2))))))
44	33	resincld 16082	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘(𝐴 − (π / 2))) ∈ ℝ)
45	44	lt0neg2d 11780	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 < (sin‘(𝐴 − (π / 2))) ↔ -(sin‘(𝐴 − (π / 2))) < 0))
46	45	anbi2d 629	. . . . 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 238	. . . 4 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((cos‘(𝐴 − (π / 2))) < 0 ∧ -(sin‘(𝐴 − (π / 2))) < 0)))
48		recn 11196	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
49		pncan3 11464	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((π / 2) + (𝐴 − (π / 2))) = 𝐴)
50	21, 48, 49	sylancr 587	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → ((π / 2) + (𝐴 − (π / 2))) = 𝐴)
51	50	fveq2d 6892	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (sin‘𝐴))
52	33	recnd 11238	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 − (π / 2)) ∈ ℂ)
53		sinhalfpip 25993	. . . . . . . 8 ⊢ ((𝐴 − (π / 2)) ∈ ℂ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘(𝐴 − (π / 2))))
54	52, 53	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘(𝐴 − (π / 2))))
55	51, 54	eqtr3d 2774	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sin‘𝐴) = (cos‘(𝐴 − (π / 2))))
56	55	breq1d 5157	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((sin‘𝐴) < 0 ↔ (cos‘(𝐴 − (π / 2))) < 0))
57	50	fveq2d 6892	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = (cos‘𝐴))
58		coshalfpip 25995	. . . . . . . 8 ⊢ ((𝐴 − (π / 2)) ∈ ℂ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = -(sin‘(𝐴 − (π / 2))))
59	52, 58	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (cos‘((π / 2) + (𝐴 − (π / 2)))) = -(sin‘(𝐴 − (π / 2))))
60	57, 59	eqtr3d 2774	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (cos‘𝐴) = -(sin‘(𝐴 − (π / 2))))
61	60	breq1d 5157	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((cos‘𝐴) < 0 ↔ -(sin‘(𝐴 − (π / 2))) < 0))
62	56, 61	anbi12d 631	. . . 4 ⊢ (𝐴 ∈ ℝ → (((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0) ↔ ((cos‘(𝐴 − (π / 2))) < 0 ∧ -(sin‘(𝐴 − (π / 2))) < 0)))
63	47, 62	sylibrd 258	. . 3 ⊢ (𝐴 ∈ ℝ → ((π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0)))
64	63	3impib 1116	. 2 ⊢ ((𝐴 ∈ ℝ ∧ π < 𝐴 ∧ 𝐴 < (3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))
65	9, 64	sylbi 216	1 ⊢ (𝐴 ∈ (π(,)(3 · (π / 2))) → ((sin‘𝐴) < 0 ∧ (cos‘𝐴) < 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 ℝ^*cxr 11243 < clt 11244 − cmin 11440 -cneg 11441 / cdiv 11867 2c2 12263 3c3 12264 (,)cioo 13320 sincsin 16003 cosccos 16004 πcpi 16006

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375

This theorem is referenced by: sincosq4sgn 26002

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