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Theorem coseq00topi 15422

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 8107	. . . . 5 ⊢ 0 ∈ ℝ
2		pire 15373	. . . . 5 ⊢ π ∈ ℝ
3	1, 2	elicc2i 10096	. . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1015	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		neghalfpire 15380	. . . . 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 15376	. . . . . . . 8 ⊢ π ∈ ℝ⁺
9		rphalfcl 9838	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
11		rpgt0 9822	. . . . . . 7 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 0 < (π / 2)
13		halfpire 15379	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14		lt0neg2 8577	. . . . . . 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 145	. . . . 5 ⊢ -(π / 2) < 0
17	16	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 0)
18	3	simp2bi 1016	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴)
19	6, 7, 4, 17, 18	ltletrd 8531	. . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴)
20	2	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ)
21		3re 9145	. . . . . 6 ⊢ 3 ∈ ℝ
22	21, 13	remulcli 8121	. . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ
23	22	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ)
24	3	simp3bi 1017	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π)
25		2div2e1 9204	. . . . . . . 8 ⊢ (2 / 2) = 1
26		2lt3 9242	. . . . . . . . 9 ⊢ 2 < 3
27		2re 9141	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
28		2pos 9162	. . . . . . . . . 10 ⊢ 0 < 2
29	27, 21, 27, 28	ltdiv1ii 9037	. . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2))
30	26, 29	mpbi 145	. . . . . . . 8 ⊢ (2 / 2) < (3 / 2)
31	25, 30	eqbrtrri 4082	. . . . . . 7 ⊢ 1 < (3 / 2)
32	21	rehalfcli 9321	. . . . . . . 8 ⊢ (3 / 2) ∈ ℝ
33		pipos 15375	. . . . . . . 8 ⊢ 0 < π
34		ltmulgt12 8973	. . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π)))
35	2, 32, 33, 34	mp3an 1350	. . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π))
36	31, 35	mpbi 145	. . . . . 6 ⊢ π < ((3 / 2) · π)
37	21	recni 8119	. . . . . . 7 ⊢ 3 ∈ ℂ
38		2cn 9142	. . . . . . . 8 ⊢ 2 ∈ ℂ
39		2ap0 9164	. . . . . . . 8 ⊢ 2 # 0
40	38, 39	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
41	2	recni 8119	. . . . . . 7 ⊢ π ∈ ℂ
42		div32ap 8800	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2)))
43	37, 40, 41, 42	mp3an 1350	. . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2))
44	36, 43	breqtri 4084	. . . . 5 ⊢ π < (3 · (π / 2))
45	44	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2)))
46	4, 20, 23, 24, 45	lelttrd 8232	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2)))
47		neghalfpirx 15381	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
48	22	rexri 8165	. . . 4 ⊢ (3 · (π / 2)) ∈ ℝ^*
49		elioo2 10078	. . . 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 426	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
51	4, 19, 46, 50	syl3anbrc 1184	. 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))))
52		coseq0q4123 15421	. 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 104 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∈ wcel 2178 class class class wbr 4059 ‘cfv 5290 (class class class)co 5967 ℂcc 7958 ℝcr 7959 0cc0 7960 1c1 7961 · cmul 7965 ℝ^*cxr 8141 < clt 8142 ≤ cle 8143 -cneg 8279 # cap 8689 / cdiv 8780 2c2 9122 3c3 9123 ℝ⁺crp 9810 (,)cioo 10045 [,]cicc 10048 cosccos 12071 πcpi 12073

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078 ax-arch 8079 ax-caucvg 8080 ax-pre-suploc 8081 ax-addf 8082 ax-mulf 8083

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-disj 4036 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-isom 5299 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-of 6181 df-1st 6249 df-2nd 6250 df-recs 6414 df-irdg 6479 df-frec 6500 df-1o 6525 df-oadd 6529 df-er 6643 df-map 6760 df-pm 6761 df-en 6851 df-dom 6852 df-fin 6853 df-sup 7112 df-inf 7113 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-5 9133 df-6 9134 df-7 9135 df-8 9136 df-9 9137 df-n0 9331 df-z 9408 df-uz 9684 df-q 9776 df-rp 9811 df-xneg 9929 df-xadd 9930 df-ioo 10049 df-ioc 10050 df-ico 10051 df-icc 10052 df-fz 10166 df-fzo 10300 df-seqfrec 10630 df-exp 10721 df-fac 10908 df-bc 10930 df-ihash 10958 df-shft 11241 df-cj 11268 df-re 11269 df-im 11270 df-rsqrt 11424 df-abs 11425 df-clim 11705 df-sumdc 11780 df-ef 12074 df-sin 12076 df-cos 12077 df-pi 12079 df-rest 13188 df-topgen 13207 df-psmet 14420 df-xmet 14421 df-met 14422 df-bl 14423 df-mopn 14424 df-top 14585 df-topon 14598 df-bases 14630 df-ntr 14683 df-cn 14775 df-cnp 14776 df-tx 14840 df-cncf 15158 df-limced 15243 df-dvap 15244

This theorem is referenced by: coseq0negpitopi 15423

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