cosq23lt0 - Intuitionistic Logic Explorer

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Theorem cosq23lt0 12914

Description: The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)

**Assertion**
Ref	Expression
cosq23lt0	⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)

**Proof of Theorem cosq23lt0**
Step	Hyp	Ref	Expression
1		elioore 9695	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ)
2	1	recnd 7794	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ)
3		sinhalfpip 12901	. . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
5		halfpire 12873	. . . . . 6 ⊢ (π / 2) ∈ ℝ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ)
7	6, 1	readdcld 7795	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ)
8		pidiv2halves 12876	. . . . 5 ⊢ ((π / 2) + (π / 2)) = π
9	5	rexri 7823	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ^*
10		3re 8794	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
11	10, 5	remulcli 7780	. . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ
12	11	rexri 7823	. . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ^*
13		elioo2 9704	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ^* ∧ (3 · (π / 2)) ∈ ℝ^*) → (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2)))))
14	9, 12, 13	mp2an 422	. . . . . . 7 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
15	14	simp2bi 997	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴)
16	6, 1, 6, 15	ltadd2dd 8184	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴))
17	8, 16	eqbrtrrid 3964	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → π < ((π / 2) + 𝐴))
18	11	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (3 · (π / 2)) ∈ ℝ)
19	14	simp3bi 998	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2)))
20	1, 18, 6, 19	ltadd2dd 8184	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2))))
21		ax-1cn 7713	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		3cn 8795	. . . . . . . 8 ⊢ 3 ∈ ℂ
23	5	recni 7778	. . . . . . . 8 ⊢ (π / 2) ∈ ℂ
24	21, 22, 23	adddiri 7777	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2)))
25		3p1e4 8855	. . . . . . . . 9 ⊢ (3 + 1) = 4
26	22, 21, 25	addcomli 7907	. . . . . . . 8 ⊢ (1 + 3) = 4
27	26	oveq1i 5784	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2))
28	23	mulid2i 7769	. . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2)
29	28	oveq1i 5784	. . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2)))
30	24, 27, 29	3eqtr3ri 2169	. . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2))
31		4cn 8798	. . . . . . 7 ⊢ 4 ∈ ℂ
32		2cn 8791	. . . . . . . 8 ⊢ 2 ∈ ℂ
33		2ap0 8813	. . . . . . . 8 ⊢ 2 # 0
34	32, 33	pm3.2i 270	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
35		picn 12868	. . . . . . 7 ⊢ π ∈ ℂ
36		div32ap 8452	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2)))
37	31, 34, 35, 36	mp3an 1315	. . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2))
38		4d2e2 8880	. . . . . . 7 ⊢ (4 / 2) = 2
39	38	oveq1i 5784	. . . . . 6 ⊢ ((4 / 2) · π) = (2 · π)
40	30, 37, 39	3eqtr2i 2166	. . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π)
41	20, 40	breqtrdi 3969	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π))
42		pire 12867	. . . . . 6 ⊢ π ∈ ℝ
43	42	rexri 7823	. . . . 5 ⊢ π ∈ ℝ^*
44		2re 8790	. . . . . . 7 ⊢ 2 ∈ ℝ
45	44, 42	remulcli 7780	. . . . . 6 ⊢ (2 · π) ∈ ℝ
46	45	rexri 7823	. . . . 5 ⊢ (2 · π) ∈ ℝ^*
47		elioo2 9704	. . . . 5 ⊢ ((π ∈ ℝ^* ∧ (2 · π) ∈ ℝ^*) → (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π))))
48	43, 46, 47	mp2an 422	. . . 4 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π)))
49	7, 17, 41, 48	syl3anbrc 1165	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π)))
50		sinq34lt0t 12912	. . 3 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) → (sin‘((π / 2) + 𝐴)) < 0)
51	49, 50	syl 14	. 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) < 0)
52	4, 51	eqbrtrrd 3952	1 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ℝ^*cxr 7799 < clt 7800 # cap 8343 / cdiv 8432 2c2 8771 3c3 8772 4c4 8773 (,)cioo 9671 sincsin 11350 cosccos 11351 πcpi 11353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 ax-pre-suploc 7741 ax-addf 7742 ax-mulf 7743

This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-disj 3907 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-map 6544 df-pm 6545 df-en 6635 df-dom 6636 df-fin 6637 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-ioo 9675 df-ioc 9676 df-ico 9677 df-icc 9678 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-fac 10472 df-bc 10494 df-ihash 10522 df-shft 10587 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 df-ef 11354 df-sin 11356 df-cos 11357 df-pi 11359 df-rest 12122 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-met 12158 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-ntr 12265 df-cn 12357 df-cnp 12358 df-tx 12422 df-cncf 12727 df-limced 12794 df-dvap 12795

This theorem is referenced by: coseq0q4123 12915 cos02pilt1 12932

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