cosq23lt0 - Intuitionistic Logic Explorer

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Theorem cosq23lt0 15547

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 10137	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ)
2	1	recnd 8198	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ)
3		sinhalfpip 15534	. . 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 15506	. . . . . 6 ⊢ (π / 2) ∈ ℝ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ)
7	6, 1	readdcld 8199	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ)
8		pidiv2halves 15509	. . . . 5 ⊢ ((π / 2) + (π / 2)) = π
9	5	rexri 8227	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ^*
10		3re 9207	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
11	10, 5	remulcli 8183	. . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ
12	11	rexri 8227	. . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ^*
13		elioo2 10146	. . . . . . . 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 1037	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴)
16	6, 1, 6, 15	ltadd2dd 8592	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴))
17	8, 16	eqbrtrrid 4122	. . . 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 1038	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2)))
20	1, 18, 6, 19	ltadd2dd 8592	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2))))
21		ax-1cn 8115	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		3cn 9208	. . . . . . . 8 ⊢ 3 ∈ ℂ
23	5	recni 8181	. . . . . . . 8 ⊢ (π / 2) ∈ ℂ
24	21, 22, 23	adddiri 8180	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2)))
25		3p1e4 9269	. . . . . . . . 9 ⊢ (3 + 1) = 4
26	22, 21, 25	addcomli 8314	. . . . . . . 8 ⊢ (1 + 3) = 4
27	26	oveq1i 6023	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2))
28	23	mullidi 8172	. . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2)
29	28	oveq1i 6023	. . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2)))
30	24, 27, 29	3eqtr3ri 2259	. . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2))
31		4cn 9211	. . . . . . 7 ⊢ 4 ∈ ℂ
32		2cn 9204	. . . . . . . 8 ⊢ 2 ∈ ℂ
33		2ap0 9226	. . . . . . . 8 ⊢ 2 # 0
34	32, 33	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
35		picn 15501	. . . . . . 7 ⊢ π ∈ ℂ
36		div32ap 8862	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2)))
37	31, 34, 35, 36	mp3an 1371	. . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2))
38		4d2e2 9294	. . . . . . 7 ⊢ (4 / 2) = 2
39	38	oveq1i 6023	. . . . . 6 ⊢ ((4 / 2) · π) = (2 · π)
40	30, 37, 39	3eqtr2i 2256	. . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π)
41	20, 40	breqtrdi 4127	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π))
42		pire 15500	. . . . . 6 ⊢ π ∈ ℝ
43	42	rexri 8227	. . . . 5 ⊢ π ∈ ℝ^*
44		2re 9203	. . . . . . 7 ⊢ 2 ∈ ℝ
45	44, 42	remulcli 8183	. . . . . 6 ⊢ (2 · π) ∈ ℝ
46	45	rexri 8227	. . . . 5 ⊢ (2 · π) ∈ ℝ^*
47		elioo2 10146	. . . . 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 1205	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π)))
50		sinq34lt0t 15545	. . 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 4110	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 1002 = wceq 1395 ∈ wcel 2200 class class class wbr 4086 ‘cfv 5324 (class class class)co 6013 ℂcc 8020 ℝcr 8021 0cc0 8022 1c1 8023 + caddc 8025 · cmul 8027 ℝ^*cxr 8203 < clt 8204 # cap 8751 / cdiv 8842 2c2 9184 3c3 9185 4c4 9186 (,)cioo 10113 sincsin 12195 cosccos 12196 πcpi 12198

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8113 ax-resscn 8114 ax-1cn 8115 ax-1re 8116 ax-icn 8117 ax-addcl 8118 ax-addrcl 8119 ax-mulcl 8120 ax-mulrcl 8121 ax-addcom 8122 ax-mulcom 8123 ax-addass 8124 ax-mulass 8125 ax-distr 8126 ax-i2m1 8127 ax-0lt1 8128 ax-1rid 8129 ax-0id 8130 ax-rnegex 8131 ax-precex 8132 ax-cnre 8133 ax-pre-ltirr 8134 ax-pre-ltwlin 8135 ax-pre-lttrn 8136 ax-pre-apti 8137 ax-pre-ltadd 8138 ax-pre-mulgt0 8139 ax-pre-mulext 8140 ax-arch 8141 ax-caucvg 8142 ax-pre-suploc 8143 ax-addf 8144 ax-mulf 8145

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-disj 4063 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-isom 5333 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-of 6230 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-frec 6552 df-1o 6577 df-oadd 6581 df-er 6697 df-map 6814 df-pm 6815 df-en 6905 df-dom 6906 df-fin 6907 df-sup 7174 df-inf 7175 df-pnf 8206 df-mnf 8207 df-xr 8208 df-ltxr 8209 df-le 8210 df-sub 8342 df-neg 8343 df-reap 8745 df-ap 8752 df-div 8843 df-inn 9134 df-2 9192 df-3 9193 df-4 9194 df-5 9195 df-6 9196 df-7 9197 df-8 9198 df-9 9199 df-n0 9393 df-z 9470 df-uz 9746 df-q 9844 df-rp 9879 df-xneg 9997 df-xadd 9998 df-ioo 10117 df-ioc 10118 df-ico 10119 df-icc 10120 df-fz 10234 df-fzo 10368 df-seqfrec 10700 df-exp 10791 df-fac 10978 df-bc 11000 df-ihash 11028 df-shft 11366 df-cj 11393 df-re 11394 df-im 11395 df-rsqrt 11549 df-abs 11550 df-clim 11830 df-sumdc 11905 df-ef 12199 df-sin 12201 df-cos 12202 df-pi 12204 df-rest 13314 df-topgen 13333 df-psmet 14547 df-xmet 14548 df-met 14549 df-bl 14550 df-mopn 14551 df-top 14712 df-topon 14725 df-bases 14757 df-ntr 14810 df-cn 14902 df-cnp 14903 df-tx 14967 df-cncf 15285 df-limced 15370 df-dvap 15371

This theorem is referenced by: coseq0q4123 15548 cos02pilt1 15565

W3C validator