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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 7899 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | pire 13357 | . . . . 5 ⊢ π ∈ ℝ | |
3 | 1, 2 | elicc2i 9875 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π)) |
4 | 3 | simp1bi 1002 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ) |
5 | neghalfpire 13364 | . . . . 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 13360 | . . . . . . . 8 ⊢ π ∈ ℝ+ | |
9 | rphalfcl 9617 | . . . . . . . 8 ⊢ (π ∈ ℝ+ → (π / 2) ∈ ℝ+) | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ+ |
11 | rpgt0 9601 | . . . . . . 7 ⊢ ((π / 2) ∈ ℝ+ → 0 < (π / 2)) | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 ⊢ 0 < (π / 2) |
13 | halfpire 13363 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ | |
14 | lt0neg2 8367 | . . . . . . 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 1003 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴) |
19 | 6, 7, 4, 17, 18 | ltletrd 8321 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴) |
20 | 2 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ) |
21 | 3re 8931 | . . . . . 6 ⊢ 3 ∈ ℝ | |
22 | 21, 13 | remulcli 7913 | . . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ |
23 | 22 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ) |
24 | 3 | simp3bi 1004 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π) |
25 | 2div2e1 8989 | . . . . . . . 8 ⊢ (2 / 2) = 1 | |
26 | 2lt3 9027 | . . . . . . . . 9 ⊢ 2 < 3 | |
27 | 2re 8927 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
28 | 2pos 8948 | . . . . . . . . . 10 ⊢ 0 < 2 | |
29 | 27, 21, 27, 28 | ltdiv1ii 8824 | . . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2)) |
30 | 26, 29 | mpbi 144 | . . . . . . . 8 ⊢ (2 / 2) < (3 / 2) |
31 | 25, 30 | eqbrtrri 4005 | . . . . . . 7 ⊢ 1 < (3 / 2) |
32 | 21 | rehalfcli 9105 | . . . . . . . 8 ⊢ (3 / 2) ∈ ℝ |
33 | pipos 13359 | . . . . . . . 8 ⊢ 0 < π | |
34 | ltmulgt12 8760 | . . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π))) | |
35 | 2, 32, 33, 34 | mp3an 1327 | . . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π)) |
36 | 31, 35 | mpbi 144 | . . . . . 6 ⊢ π < ((3 / 2) · π) |
37 | 21 | recni 7911 | . . . . . . 7 ⊢ 3 ∈ ℂ |
38 | 2cn 8928 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
39 | 2ap0 8950 | . . . . . . . 8 ⊢ 2 # 0 | |
40 | 38, 39 | pm3.2i 270 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0) |
41 | 2 | recni 7911 | . . . . . . 7 ⊢ π ∈ ℂ |
42 | div32ap 8588 | . . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2))) | |
43 | 37, 40, 41, 42 | mp3an 1327 | . . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2)) |
44 | 36, 43 | breqtri 4007 | . . . . 5 ⊢ π < (3 · (π / 2)) |
45 | 44 | a1i 9 | . . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2))) |
46 | 4, 20, 23, 24, 45 | lelttrd 8023 | . . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2))) |
47 | neghalfpirx 13365 | . . . 4 ⊢ -(π / 2) ∈ ℝ* | |
48 | 22 | rexri 7956 | . . . 4 ⊢ (3 · (π / 2)) ∈ ℝ* |
49 | elioo2 9857 | . . . 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 1171 | . 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2)))) |
52 | coseq0q4123 13405 | . 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 968 = wceq 1343 ∈ wcel 2136 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 ℝcr 7752 0cc0 7753 1c1 7754 · cmul 7758 ℝ*cxr 7932 < clt 7933 ≤ cle 7934 -cneg 8070 # cap 8479 / cdiv 8568 2c2 8908 3c3 8909 ℝ+crp 9589 (,)cioo 9824 [,]cicc 9827 cosccos 11586 πcpi 11588 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ioc 9829 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-sin 11591 df-cos 11592 df-pi 11594 df-rest 12558 df-topgen 12577 df-psmet 12637 df-xmet 12638 df-met 12639 df-bl 12640 df-mopn 12641 df-top 12646 df-topon 12659 df-bases 12691 df-ntr 12746 df-cn 12838 df-cnp 12839 df-tx 12903 df-cncf 13208 df-limced 13275 df-dvap 13276 |
This theorem is referenced by: coseq0negpitopi 13407 |
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