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Theorem coseq00topi 15986

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 8326	. . . . 5 ⊢ 0 ∈ ℝ
2		pire 15937	. . . . 5 ⊢ π ∈ ℝ
3	1, 2	elicc2i 10351	. . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1043	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		neghalfpire 15944	. . . . 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 15940	. . . . . . . 8 ⊢ π ∈ ℝ⁺
9		rphalfcl 10092	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
11		rpgt0 10076	. . . . . . 7 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 0 < (π / 2)
13		halfpire 15943	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14		lt0neg2 8798	. . . . . . 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 1044	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴)
19	6, 7, 4, 17, 18	ltletrd 8752	. . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴)
20	2	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ)
21		3re 9380	. . . . . 6 ⊢ 3 ∈ ℝ
22	21, 13	remulcli 8340	. . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ
23	22	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ)
24	3	simp3bi 1045	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π)
25		2div2e1 9439	. . . . . . . 8 ⊢ (2 / 2) = 1
26		2lt3 9479	. . . . . . . . 9 ⊢ 2 < 3
27		2re 9376	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
28		2pos 9397	. . . . . . . . . 10 ⊢ 0 < 2
29	27, 21, 27, 28	ltdiv1ii 9261	. . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2))
30	26, 29	mpbi 145	. . . . . . . 8 ⊢ (2 / 2) < (3 / 2)
31	25, 30	eqbrtrri 4153	. . . . . . 7 ⊢ 1 < (3 / 2)
32	21	rehalfcli 9558	. . . . . . . 8 ⊢ (3 / 2) ∈ ℝ
33		pipos 15939	. . . . . . . 8 ⊢ 0 < π
34		ltmulgt12 9197	. . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π)))
35	2, 32, 33, 34	mp3an 1378	. . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π))
36	31, 35	mpbi 145	. . . . . 6 ⊢ π < ((3 / 2) · π)
37	21	recni 8338	. . . . . . 7 ⊢ 3 ∈ ℂ
38		2cn 9377	. . . . . . . 8 ⊢ 2 ∈ ℂ
39		2ap0 9399	. . . . . . . 8 ⊢ 2 # 0
40	38, 39	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
41	2	recni 8338	. . . . . . 7 ⊢ π ∈ ℂ
42		div32ap 9024	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2)))
43	37, 40, 41, 42	mp3an 1378	. . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2))
44	36, 43	breqtri 4155	. . . . 5 ⊢ π < (3 · (π / 2))
45	44	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2)))
46	4, 20, 23, 24, 45	lelttrd 8452	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2)))
47		neghalfpirx 15945	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
48	22	rexri 8383	. . . 4 ⊢ (3 · (π / 2)) ∈ ℝ^*
49		elioo2 10333	. . . 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 430	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
51	4, 19, 46, 50	syl3anbrc 1212	. 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))))
52		coseq0q4123 15985	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 ℂcc 8177 ℝcr 8178 0cc0 8179 1c1 8180 · cmul 8184 ℝ^*cxr 8359 < clt 8360 ≤ cle 8361 -cneg 8499 # cap 8911 / cdiv 9004 2c2 9357 3c3 9358 ℝ⁺crp 10064 (,)cioo 10300 [,]cicc 10303 cosccos 12428 πcpi 12430

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299 ax-pre-suploc 8300 ax-addf 8301 ax-mulf 8302

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-disj 4107 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-isom 5386 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-of 6302 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-oadd 6691 df-er 6807 df-map 6924 df-pm 6925 df-en 7023 df-dom 7024 df-fin 7025 df-sup 7324 df-inf 7325 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8500 df-neg 8501 df-reap 8905 df-ap 8912 df-div 9005 df-inn 9307 df-2 9365 df-3 9366 df-4 9367 df-5 9368 df-6 9369 df-7 9370 df-8 9371 df-9 9372 df-n0 9568 df-z 9649 df-uz 9931 df-q 10029 df-rp 10065 df-xneg 10184 df-xadd 10185 df-ioo 10304 df-ioc 10305 df-ico 10306 df-icc 10307 df-fz 10422 df-fzo 10560 df-seqfrec 10898 df-exp 10989 df-fac 11178 df-bc 11200 df-ihash 11229 df-shft 11594 df-cj 11621 df-re 11622 df-im 11623 df-rsqrt 11778 df-abs 11779 df-clim 12061 df-sumdc 12136 df-ef 12431 df-sin 12433 df-cos 12434 df-pi 12436 df-rest 13644 df-topgen 13663 df-psmet 14929 df-xmet 14930 df-met 14931 df-bl 14932 df-mopn 14933 df-top 15148 df-topon 15161 df-bases 15193 df-ntr 15246 df-cn 15338 df-cnp 15339 df-tx 15403 df-cncf 15721 df-limced 15806 df-dvap 15807

This theorem is used by: coseq0negpitopi 15987

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