coseq00topi - Intuitionistic Logic Explorer

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Theorem coseq00topi 13550

Description: Location of the zeroes of cosine in (0[,]π). (Contributed by David Moews, 28-Feb-2017.)

**Assertion**
Ref	Expression
coseq00topi	⊢ (𝐴 ∈ (0[,]π) → ((cos‘𝐴) = 0 ↔ 𝐴 = (π / 2)))

**Proof of Theorem coseq00topi**
Step	Hyp	Ref	Expression
1		0re 7920	. . . . 5 ⊢ 0 ∈ ℝ
2		pire 13501	. . . . 5 ⊢ π ∈ ℝ
3	1, 2	elicc2i 9896	. . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1007	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		neghalfpire 13508	. . . . 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 13504	. . . . . . . 8 ⊢ π ∈ ℝ⁺
9		rphalfcl 9638	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
11		rpgt0 9622	. . . . . . 7 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 0 < (π / 2)
13		halfpire 13507	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14		lt0neg2 8388	. . . . . . 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 1008	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴)
19	6, 7, 4, 17, 18	ltletrd 8342	. . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴)
20	2	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ)
21		3re 8952	. . . . . 6 ⊢ 3 ∈ ℝ
22	21, 13	remulcli 7934	. . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ
23	22	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ)
24	3	simp3bi 1009	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π)
25		2div2e1 9010	. . . . . . . 8 ⊢ (2 / 2) = 1
26		2lt3 9048	. . . . . . . . 9 ⊢ 2 < 3
27		2re 8948	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
28		2pos 8969	. . . . . . . . . 10 ⊢ 0 < 2
29	27, 21, 27, 28	ltdiv1ii 8845	. . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2))
30	26, 29	mpbi 144	. . . . . . . 8 ⊢ (2 / 2) < (3 / 2)
31	25, 30	eqbrtrri 4012	. . . . . . 7 ⊢ 1 < (3 / 2)
32	21	rehalfcli 9126	. . . . . . . 8 ⊢ (3 / 2) ∈ ℝ
33		pipos 13503	. . . . . . . 8 ⊢ 0 < π
34		ltmulgt12 8781	. . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π)))
35	2, 32, 33, 34	mp3an 1332	. . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π))
36	31, 35	mpbi 144	. . . . . 6 ⊢ π < ((3 / 2) · π)
37	21	recni 7932	. . . . . . 7 ⊢ 3 ∈ ℂ
38		2cn 8949	. . . . . . . 8 ⊢ 2 ∈ ℂ
39		2ap0 8971	. . . . . . . 8 ⊢ 2 # 0
40	38, 39	pm3.2i 270	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
41	2	recni 7932	. . . . . . 7 ⊢ π ∈ ℂ
42		div32ap 8609	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2)))
43	37, 40, 41, 42	mp3an 1332	. . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2))
44	36, 43	breqtri 4014	. . . . 5 ⊢ π < (3 · (π / 2))
45	44	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2)))
46	4, 20, 23, 24, 45	lelttrd 8044	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2)))
47		neghalfpirx 13509	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
48	22	rexri 7977	. . . 4 ⊢ (3 · (π / 2)) ∈ ℝ^*
49		elioo2 9878	. . . 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 424	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
51	4, 19, 46, 50	syl3anbrc 1176	. 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))))
52		coseq0q4123 13549	. 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 973 = wceq 1348 ∈ wcel 2141 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 · cmul 7779 ℝ^*cxr 7953 < clt 7954 ≤ cle 7955 -cneg 8091 # cap 8500 / cdiv 8589 2c2 8929 3c3 8930 ℝ⁺crp 9610 (,)cioo 9845 [,]cicc 9848 cosccos 11608 πcpi 11610

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 ax-pre-suploc 7895 ax-addf 7896 ax-mulf 7897

This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-disj 3967 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-of 6061 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-map 6628 df-pm 6629 df-en 6719 df-dom 6720 df-fin 6721 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-ioo 9849 df-ioc 9850 df-ico 9851 df-icc 9852 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-fac 10660 df-bc 10682 df-ihash 10710 df-shft 10779 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 df-ef 11611 df-sin 11613 df-cos 11614 df-pi 11616 df-rest 12581 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-topon 12803 df-bases 12835 df-ntr 12890 df-cn 12982 df-cnp 12983 df-tx 13047 df-cncf 13352 df-limced 13419 df-dvap 13420

This theorem is referenced by: coseq0negpitopi 13551

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