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Mirrors > Home > ILE Home > Th. List > cosq23lt0 | GIF version |
Description: The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.) |
Ref | Expression |
---|---|
cosq23lt0 | ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0) |
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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