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Theorem coseq00topi 15307

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 8072	. . . . 5 ⊢ 0 ∈ ℝ
2		pire 15258	. . . . 5 ⊢ π ∈ ℝ
3	1, 2	elicc2i 10061	. . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1015	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		neghalfpire 15265	. . . . 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 15261	. . . . . . . 8 ⊢ π ∈ ℝ⁺
9		rphalfcl 9803	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
11		rpgt0 9787	. . . . . . 7 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 0 < (π / 2)
13		halfpire 15264	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14		lt0neg2 8542	. . . . . . 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 8496	. . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴)
20	2	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ)
21		3re 9110	. . . . . 6 ⊢ 3 ∈ ℝ
22	21, 13	remulcli 8086	. . . . 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 9169	. . . . . . . 8 ⊢ (2 / 2) = 1
26		2lt3 9207	. . . . . . . . 9 ⊢ 2 < 3
27		2re 9106	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
28		2pos 9127	. . . . . . . . . 10 ⊢ 0 < 2
29	27, 21, 27, 28	ltdiv1ii 9002	. . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2))
30	26, 29	mpbi 145	. . . . . . . 8 ⊢ (2 / 2) < (3 / 2)
31	25, 30	eqbrtrri 4067	. . . . . . 7 ⊢ 1 < (3 / 2)
32	21	rehalfcli 9286	. . . . . . . 8 ⊢ (3 / 2) ∈ ℝ
33		pipos 15260	. . . . . . . 8 ⊢ 0 < π
34		ltmulgt12 8938	. . . . . . . 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 8084	. . . . . . 7 ⊢ 3 ∈ ℂ
38		2cn 9107	. . . . . . . 8 ⊢ 2 ∈ ℂ
39		2ap0 9129	. . . . . . . 8 ⊢ 2 # 0
40	38, 39	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
41	2	recni 8084	. . . . . . 7 ⊢ π ∈ ℂ
42		div32ap 8765	. . . . . . 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 4069	. . . . 5 ⊢ π < (3 · (π / 2))
45	44	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2)))
46	4, 20, 23, 24, 45	lelttrd 8197	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2)))
47		neghalfpirx 15266	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
48	22	rexri 8130	. . . 4 ⊢ (3 · (π / 2)) ∈ ℝ^*
49		elioo2 10043	. . . 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 15306	. 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 2176 class class class wbr 4044 ‘cfv 5271 (class class class)co 5944 ℂcc 7923 ℝcr 7924 0cc0 7925 1c1 7926 · cmul 7930 ℝ^*cxr 8106 < clt 8107 ≤ cle 8108 -cneg 8244 # cap 8654 / cdiv 8745 2c2 9087 3c3 9088 ℝ⁺crp 9775 (,)cioo 10010 [,]cicc 10013 cosccos 11956 πcpi 11958

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045 ax-pre-suploc 8046 ax-addf 8047 ax-mulf 8048

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-disj 4022 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-of 6158 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-oadd 6506 df-er 6620 df-map 6737 df-pm 6738 df-en 6828 df-dom 6829 df-fin 6830 df-sup 7086 df-inf 7087 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-xneg 9894 df-xadd 9895 df-ioo 10014 df-ioc 10015 df-ico 10016 df-icc 10017 df-fz 10131 df-fzo 10265 df-seqfrec 10593 df-exp 10684 df-fac 10871 df-bc 10893 df-ihash 10921 df-shft 11126 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-sumdc 11665 df-ef 11959 df-sin 11961 df-cos 11962 df-pi 11964 df-rest 13073 df-topgen 13092 df-psmet 14305 df-xmet 14306 df-met 14307 df-bl 14308 df-mopn 14309 df-top 14470 df-topon 14483 df-bases 14515 df-ntr 14568 df-cn 14660 df-cnp 14661 df-tx 14725 df-cncf 15043 df-limced 15128 df-dvap 15129

This theorem is referenced by: coseq0negpitopi 15308

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