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Mirrors > Home > ILE Home > Th. List > coseq00topi | GIF version |
Description: Location of the zeroes of cosine in (0[,]π). (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
coseq00topi | ⊢ (𝐴 ∈ (0[,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7790 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | pire 12915 | . . . . 5 ⊢ π ∈ ℝ | |
3 | 1, 2 | elicc2i 9752 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π)) |
4 | 3 | simp1bi 997 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ) |
5 | neghalfpire 12922 | . . . . 5 ⊢ -(π / 2) ∈ ℝ | |
6 | 5 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) ∈ ℝ) |
7 | 1 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ∈ ℝ) |
8 | pirp 12918 | . . . . . . . 8 ⊢ π ∈ ℝ+ | |
9 | rphalfcl 9498 | . . . . . . . 8 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ+ |
11 | rpgt0 9482 | . . . . . . 7 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 2) |
13 | halfpire 12921 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ | |
14 | lt0neg2 8255 | . . . . . . 7 ⊢ ((π / 2) ∈ ℝ → (0 < (π / 2) ↔ -(π / 2) < 0)) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (0 < (π / 2) ↔ -(π / 2) < 0) |
16 | 12, 15 | mpbi 144 | . . . . 5 ⊢ -(π / 2) < 0 |
17 | 16 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 0) |
18 | 3 | simp2bi 998 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴) |
19 | 6, 7, 4, 17, 18 | ltletrd 8209 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴) |
20 | 2 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ) |
21 | 3re 8818 | . . . . . 6 ⊢ 3 ∈ ℝ | |
22 | 21, 13 | remulcli 7804 | . . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ |
23 | 22 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ) |
24 | 3 | simp3bi 999 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π) |
25 | 2div2e1 8876 | . . . . . . . 8 ⊢ (2 / 2) = 1 | |
26 | 2lt3 8914 | . . . . . . . . 9 ⊢ 2 < 3 | |
27 | 2re 8814 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
28 | 2pos 8835 | . . . . . . . . . 10 ⊢ 0 < 2 | |
29 | 27, 21, 27, 28 | ltdiv1ii 8711 | . . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2)) |
30 | 26, 29 | mpbi 144 | . . . . . . . 8 ⊢ (2 / 2) < (3 / 2) |
31 | 25, 30 | eqbrtrri 3959 | . . . . . . 7 ⊢ 1 < (3 / 2) |
32 | 21 | rehalfcli 8992 | . . . . . . . 8 ⊢ (3 / 2) ∈ ℝ |
33 | pipos 12917 | . . . . . . . 8 ⊢ 0 < π | |
34 | ltmulgt12 8647 | . . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π))) | |
35 | 2, 32, 33, 34 | mp3an 1316 | . . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π)) |
36 | 31, 35 | mpbi 144 | . . . . . 6 ⊢ π < ((3 / 2) · π) |
37 | 21 | recni 7802 | . . . . . . 7 ⊢ 3 ∈ ℂ |
38 | 2cn 8815 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
39 | 2ap0 8837 | . . . . . . . 8 ⊢ 2 # 0 | |
40 | 38, 39 | pm3.2i 270 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0) |
41 | 2 | recni 7802 | . . . . . . 7 ⊢ π ∈ ℂ |
42 | div32ap 8476 | . . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2))) | |
43 | 37, 40, 41, 42 | mp3an 1316 | . . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2)) |
44 | 36, 43 | breqtri 3961 | . . . . 5 ⊢ π < (3 · (π / 2)) |
45 | 44 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2))) |
46 | 4, 20, 23, 24, 45 | lelttrd 7911 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2))) |
47 | neghalfpirx 12923 | . . . 4 ⊢ -(π / 2) ∈ ℝ* | |
48 | 22 | rexri 7847 | . . . 4 ⊢ (3 · (π / 2)) ∈ ℝ* |
49 | elioo2 9734 | . . . 4 ⊢ ((-(π / 2) ∈ ℝ* ∧ (3 · (π / 2)) ∈ ℝ*) → (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))) | |
50 | 47, 48, 49 | mp2an 423 | . . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2)))) |
51 | 4, 19, 46, 50 | syl3anbrc 1166 | . 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2)))) |
52 | coseq0q4123 12963 | . 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2))) | |
53 | 51, 52 | syl 14 | 1 ⊢ (𝐴 ∈ (0[,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 1c1 7645 · cmul 7649 ℝ*cxr 7823 < clt 7824 ≤ cle 7825 -cneg 7958 # cap 8367 / cdiv 8456 2c2 8795 3c3 8796 ℝ+crp 9470 (,)cioo 9701 [,]cicc 9704 cosccos 11388 πcpi 11390 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 ax-pre-suploc 7765 ax-addf 7766 ax-mulf 7767 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-disj 3915 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-isom 5140 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-of 5990 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-frec 6296 df-1o 6321 df-oadd 6325 df-er 6437 df-map 6552 df-pm 6553 df-en 6643 df-dom 6644 df-fin 6645 df-sup 6879 df-inf 6880 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-xneg 9589 df-xadd 9590 df-ioo 9705 df-ioc 9706 df-ico 9707 df-icc 9708 df-fz 9822 df-fzo 9951 df-seqfrec 10250 df-exp 10324 df-fac 10504 df-bc 10526 df-ihash 10554 df-shft 10619 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-clim 11080 df-sumdc 11155 df-ef 11391 df-sin 11393 df-cos 11394 df-pi 11396 df-rest 12161 df-topgen 12180 df-psmet 12195 df-xmet 12196 df-met 12197 df-bl 12198 df-mopn 12199 df-top 12204 df-topon 12217 df-bases 12249 df-ntr 12304 df-cn 12396 df-cnp 12397 df-tx 12461 df-cncf 12766 df-limced 12833 df-dvap 12834 |
This theorem is referenced by: coseq0negpitopi 12965 |
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