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Mirrors > Home > ILE Home > Th. List > cosq14gt0 | GIF version |
Description: The cosine of a number strictly between -π / 2 and π / 2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.) |
Ref | Expression |
---|---|
cosq14gt0 | ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfpire 12873 | . . . . 5 ⊢ (π / 2) ∈ ℝ | |
2 | elioore 9695 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ) | |
3 | resubcl 8026 | . . . . 5 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ) | |
4 | 1, 2, 3 | sylancr 410 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ) |
5 | neghalfpirx 12875 | . . . . . . 7 ⊢ -(π / 2) ∈ ℝ* | |
6 | 1 | rexri 7823 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ* |
7 | elioo2 9704 | . . . . . . 7 ⊢ ((-(π / 2) ∈ ℝ* ∧ (π / 2) ∈ ℝ*) → (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))) | |
8 | 5, 6, 7 | mp2an 422 | . . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2))) |
9 | 8 | simp3bi 998 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2)) |
10 | posdif 8217 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴))) | |
11 | 2, 1, 10 | sylancl 409 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴))) |
12 | 9, 11 | mpbid 146 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < ((π / 2) − 𝐴)) |
13 | picn 12868 | . . . . . . . 8 ⊢ π ∈ ℂ | |
14 | halfcl 8946 | . . . . . . . 8 ⊢ (π ∈ ℂ → (π / 2) ∈ ℂ) | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (π / 2) ∈ ℂ |
16 | 15 | negcli 8030 | . . . . . . 7 ⊢ -(π / 2) ∈ ℂ |
17 | 13, 15 | negsubi 8040 | . . . . . . . 8 ⊢ (π + -(π / 2)) = (π − (π / 2)) |
18 | pidiv2halves 12876 | . . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π | |
19 | 13, 15, 15, 18 | subaddrii 8051 | . . . . . . . 8 ⊢ (π − (π / 2)) = (π / 2) |
20 | 17, 19 | eqtri 2160 | . . . . . . 7 ⊢ (π + -(π / 2)) = (π / 2) |
21 | 15, 13, 16, 20 | subaddrii 8051 | . . . . . 6 ⊢ ((π / 2) − π) = -(π / 2) |
22 | 8 | simp2bi 997 | . . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴) |
23 | 21, 22 | eqbrtrid 3963 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴) |
24 | pire 12867 | . . . . . . 7 ⊢ π ∈ ℝ | |
25 | ltsub23 8204 | . . . . . . 7 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ π ∈ ℝ) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴)) | |
26 | 1, 24, 25 | mp3an13 1306 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴)) |
27 | 2, 26 | syl 14 | . . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴)) |
28 | 23, 27 | mpbird 166 | . . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π) |
29 | 0xr 7812 | . . . . 5 ⊢ 0 ∈ ℝ* | |
30 | 24 | rexri 7823 | . . . . 5 ⊢ π ∈ ℝ* |
31 | elioo2 9704 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ π ∈ ℝ*) → (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π))) | |
32 | 29, 30, 31 | mp2an 422 | . . . 4 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π)) |
33 | 4, 12, 28, 32 | syl3anbrc 1165 | . . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ (0(,)π)) |
34 | sinq12gt0 12911 | . . 3 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) → 0 < (sin‘((π / 2) − 𝐴))) | |
35 | 33, 34 | syl 14 | . 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (sin‘((π / 2) − 𝐴))) |
36 | 2 | recnd 7794 | . . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ) |
37 | sinhalfpim 12902 | . . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴)) | |
38 | 36, 37 | syl 14 | . 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴)) |
39 | 35, 38 | breqtrd 3954 | 1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ 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 + caddc 7623 ℝ*cxr 7799 < clt 7800 − cmin 7933 -cneg 7934 / cdiv 8432 2c2 8771 (,)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 |
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