cosq23lt0 - Intuitionistic Logic Explorer

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Theorem cosq23lt0 15627

Description: The cosine of a number in the second and third quadrants is negative. (Contributed by Jim Kingdon, 14-Mar-2024.)

**Assertion**
Ref	Expression
cosq23lt0	⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)

**Proof of Theorem cosq23lt0**
Step	Hyp	Ref	Expression
1		elioore 10191	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ)
2	1	recnd 8250	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ)
3		sinhalfpip 15614	. . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
4	2, 3	syl 14	. 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) = (cos‘𝐴))
5		halfpire 15586	. . . . . 6 ⊢ (π / 2) ∈ ℝ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ)
7	6, 1	readdcld 8251	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ)
8		pidiv2halves 15589	. . . . 5 ⊢ ((π / 2) + (π / 2)) = π
9	5	rexri 8279	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ^*
10		3re 9259	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
11	10, 5	remulcli 8236	. . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ
12	11	rexri 8279	. . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ^*
13		elioo2 10200	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ^* ∧ (3 · (π / 2)) ∈ ℝ^*) → (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2)))))
14	9, 12, 13	mp2an 426	. . . . . . 7 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
15	14	simp2bi 1040	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴)
16	6, 1, 6, 15	ltadd2dd 8644	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴))
17	8, 16	eqbrtrrid 4129	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → π < ((π / 2) + 𝐴))
18	11	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (3 · (π / 2)) ∈ ℝ)
19	14	simp3bi 1041	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2)))
20	1, 18, 6, 19	ltadd2dd 8644	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2))))
21		ax-1cn 8168	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		3cn 9260	. . . . . . . 8 ⊢ 3 ∈ ℂ
23	5	recni 8234	. . . . . . . 8 ⊢ (π / 2) ∈ ℂ
24	21, 22, 23	adddiri 8233	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2)))
25		3p1e4 9321	. . . . . . . . 9 ⊢ (3 + 1) = 4
26	22, 21, 25	addcomli 8366	. . . . . . . 8 ⊢ (1 + 3) = 4
27	26	oveq1i 6038	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2))
28	23	mullidi 8225	. . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2)
29	28	oveq1i 6038	. . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2)))
30	24, 27, 29	3eqtr3ri 2261	. . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2))
31		4cn 9263	. . . . . . 7 ⊢ 4 ∈ ℂ
32		2cn 9256	. . . . . . . 8 ⊢ 2 ∈ ℂ
33		2ap0 9278	. . . . . . . 8 ⊢ 2 # 0
34	32, 33	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
35		picn 15581	. . . . . . 7 ⊢ π ∈ ℂ
36		div32ap 8914	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2)))
37	31, 34, 35, 36	mp3an 1374	. . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2))
38		4d2e2 9346	. . . . . . 7 ⊢ (4 / 2) = 2
39	38	oveq1i 6038	. . . . . 6 ⊢ ((4 / 2) · π) = (2 · π)
40	30, 37, 39	3eqtr2i 2258	. . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π)
41	20, 40	breqtrdi 4134	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π))
42		pire 15580	. . . . . 6 ⊢ π ∈ ℝ
43	42	rexri 8279	. . . . 5 ⊢ π ∈ ℝ^*
44		2re 9255	. . . . . . 7 ⊢ 2 ∈ ℝ
45	44, 42	remulcli 8236	. . . . . 6 ⊢ (2 · π) ∈ ℝ
46	45	rexri 8279	. . . . 5 ⊢ (2 · π) ∈ ℝ^*
47		elioo2 10200	. . . . 5 ⊢ ((π ∈ ℝ^* ∧ (2 · π) ∈ ℝ^*) → (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π))))
48	43, 46, 47	mp2an 426	. . . 4 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) ↔ (((π / 2) + 𝐴) ∈ ℝ ∧ π < ((π / 2) + 𝐴) ∧ ((π / 2) + 𝐴) < (2 · π)))
49	7, 17, 41, 48	syl3anbrc 1208	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π)))
50		sinq34lt0t 15625	. . 3 ⊢ (((π / 2) + 𝐴) ∈ (π(,)(2 · π)) → (sin‘((π / 2) + 𝐴)) < 0)
51	49, 50	syl 14	. 2 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (sin‘((π / 2) + 𝐴)) < 0)
52	4, 51	eqbrtrrd 4117	1 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2202 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 ℝcr 8074 0cc0 8075 1c1 8076 + caddc 8078 · cmul 8080 ℝ^*cxr 8255 < clt 8256 # cap 8803 / cdiv 8894 2c2 9236 3c3 9237 4c4 9238 (,)cioo 10167 sincsin 12268 cosccos 12269 πcpi 12271

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195 ax-pre-suploc 8196 ax-addf 8197 ax-mulf 8198

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-disj 4070 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-isom 5342 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-of 6244 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-oadd 6629 df-er 6745 df-map 6862 df-pm 6863 df-en 6953 df-dom 6954 df-fin 6955 df-sup 7226 df-inf 7227 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248 df-7 9249 df-8 9250 df-9 9251 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-xneg 10051 df-xadd 10052 df-ioo 10171 df-ioc 10172 df-ico 10173 df-icc 10174 df-fz 10289 df-fzo 10423 df-seqfrec 10756 df-exp 10847 df-fac 11034 df-bc 11056 df-ihash 11084 df-shft 11438 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-clim 11902 df-sumdc 11977 df-ef 12272 df-sin 12274 df-cos 12275 df-pi 12277 df-rest 13387 df-topgen 13406 df-psmet 14622 df-xmet 14623 df-met 14624 df-bl 14625 df-mopn 14626 df-top 14792 df-topon 14805 df-bases 14837 df-ntr 14890 df-cn 14982 df-cnp 14983 df-tx 15047 df-cncf 15365 df-limced 15450 df-dvap 15451

This theorem is referenced by: coseq0q4123 15628 cos02pilt1 15645

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