coseq00topi - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > coseq00topi		GIF version

Theorem coseq00topi 15578

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 8179	. . . . 5 ⊢ 0 ∈ ℝ
2		pire 15529	. . . . 5 ⊢ π ∈ ℝ
3	1, 2	elicc2i 10174	. . . 4 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1038	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		neghalfpire 15536	. . . . 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 15532	. . . . . . . 8 ⊢ π ∈ ℝ⁺
9		rphalfcl 9916	. . . . . . . 8 ⊢ (π ∈ ℝ⁺ → (π / 2) ∈ ℝ⁺)
10	8, 9	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℝ⁺
11		rpgt0 9900	. . . . . . 7 ⊢ ((π / 2) ∈ ℝ⁺ → 0 < (π / 2))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ 0 < (π / 2)
13		halfpire 15535	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
14		lt0neg2 8649	. . . . . . 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 1039	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴)
19	6, 7, 4, 17, 18	ltletrd 8603	. . 3 ⊢ (𝐴 ∈ (0[,]π) → -(π / 2) < 𝐴)
20	2	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π ∈ ℝ)
21		3re 9217	. . . . . 6 ⊢ 3 ∈ ℝ
22	21, 13	remulcli 8193	. . . . 5 ⊢ (3 · (π / 2)) ∈ ℝ
23	22	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → (3 · (π / 2)) ∈ ℝ)
24	3	simp3bi 1040	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π)
25		2div2e1 9276	. . . . . . . 8 ⊢ (2 / 2) = 1
26		2lt3 9314	. . . . . . . . 9 ⊢ 2 < 3
27		2re 9213	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
28		2pos 9234	. . . . . . . . . 10 ⊢ 0 < 2
29	27, 21, 27, 28	ltdiv1ii 9109	. . . . . . . . 9 ⊢ (2 < 3 ↔ (2 / 2) < (3 / 2))
30	26, 29	mpbi 145	. . . . . . . 8 ⊢ (2 / 2) < (3 / 2)
31	25, 30	eqbrtrri 4111	. . . . . . 7 ⊢ 1 < (3 / 2)
32	21	rehalfcli 9393	. . . . . . . 8 ⊢ (3 / 2) ∈ ℝ
33		pipos 15531	. . . . . . . 8 ⊢ 0 < π
34		ltmulgt12 9045	. . . . . . . 8 ⊢ ((π ∈ ℝ ∧ (3 / 2) ∈ ℝ ∧ 0 < π) → (1 < (3 / 2) ↔ π < ((3 / 2) · π)))
35	2, 32, 33, 34	mp3an 1373	. . . . . . 7 ⊢ (1 < (3 / 2) ↔ π < ((3 / 2) · π))
36	31, 35	mpbi 145	. . . . . 6 ⊢ π < ((3 / 2) · π)
37	21	recni 8191	. . . . . . 7 ⊢ 3 ∈ ℂ
38		2cn 9214	. . . . . . . 8 ⊢ 2 ∈ ℂ
39		2ap0 9236	. . . . . . . 8 ⊢ 2 # 0
40	38, 39	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
41	2	recni 8191	. . . . . . 7 ⊢ π ∈ ℂ
42		div32ap 8872	. . . . . . 7 ⊢ ((3 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((3 / 2) · π) = (3 · (π / 2)))
43	37, 40, 41, 42	mp3an 1373	. . . . . 6 ⊢ ((3 / 2) · π) = (3 · (π / 2))
44	36, 43	breqtri 4113	. . . . 5 ⊢ π < (3 · (π / 2))
45	44	a1i 9	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → π < (3 · (π / 2)))
46	4, 20, 23, 24, 45	lelttrd 8304	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 < (3 · (π / 2)))
47		neghalfpirx 15537	. . . 4 ⊢ -(π / 2) ∈ ℝ^*
48	22	rexri 8237	. . . 4 ⊢ (3 · (π / 2)) ∈ ℝ^*
49		elioo2 10156	. . . 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 1207	. 2 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ (-(π / 2)(,)(3 · (π / 2))))
52		coseq0q4123 15577	. 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 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 ℂcc 8030 ℝcr 8031 0cc0 8032 1c1 8033 · cmul 8037 ℝ^*cxr 8213 < clt 8214 ≤ cle 8215 -cneg 8351 # cap 8761 / cdiv 8852 2c2 9194 3c3 9195 ℝ⁺crp 9888 (,)cioo 10123 [,]cicc 10126 cosccos 12224 πcpi 12226

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-5 9205 df-6 9206 df-7 9207 df-8 9208 df-9 9209 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ioc 10128 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11393 df-cj 11420 df-re 11421 df-im 11422 df-rsqrt 11576 df-abs 11577 df-clim 11857 df-sumdc 11932 df-ef 12227 df-sin 12229 df-cos 12230 df-pi 12232 df-rest 13342 df-topgen 13361 df-psmet 14576 df-xmet 14577 df-met 14578 df-bl 14579 df-mopn 14580 df-top 14741 df-topon 14754 df-bases 14786 df-ntr 14839 df-cn 14931 df-cnp 14932 df-tx 14996 df-cncf 15314 df-limced 15399 df-dvap 15400

This theorem is referenced by: coseq0negpitopi 15579

W3C validator