cosq23lt0 - Intuitionistic Logic Explorer

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Theorem cosq23lt0 15857

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 10293	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ)
2	1	recnd 8344	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ)
3		sinhalfpip 15844	. . 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 15816	. . . . . 6 ⊢ (π / 2) ∈ ℝ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ)
7	6, 1	readdcld 8345	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ)
8		pidiv2halves 15819	. . . . 5 ⊢ ((π / 2) + (π / 2)) = π
9	5	rexri 8373	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ^*
10		3re 9357	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
11	10, 5	remulcli 8330	. . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ
12	11	rexri 8373	. . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ^*
13		elioo2 10302	. . . . . . . 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 430	. . . . . . 7 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
15	14	simp2bi 1044	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴)
16	6, 1, 6, 15	ltadd2dd 8740	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴))
17	8, 16	eqbrtrrid 4161	. . . 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 1045	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2)))
20	1, 18, 6, 19	ltadd2dd 8740	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2))))
21		ax-1cn 8262	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		3cn 9358	. . . . . . . 8 ⊢ 3 ∈ ℂ
23	5	recni 8328	. . . . . . . 8 ⊢ (π / 2) ∈ ℂ
24	21, 22, 23	adddiri 8327	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2)))
25		3p1e4 9419	. . . . . . . . 9 ⊢ (3 + 1) = 4
26	22, 21, 25	addcomli 8461	. . . . . . . 8 ⊢ (1 + 3) = 4
27	26	oveq1i 6085	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2))
28	23	mullidi 8319	. . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2)
29	28	oveq1i 6085	. . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2)))
30	24, 27, 29	3eqtr3ri 2268	. . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2))
31		4cn 9361	. . . . . . 7 ⊢ 4 ∈ ℂ
32		2cn 9354	. . . . . . . 8 ⊢ 2 ∈ ℂ
33		2ap0 9376	. . . . . . . 8 ⊢ 2 # 0
34	32, 33	pm3.2i 272	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
35		picn 15811	. . . . . . 7 ⊢ π ∈ ℂ
36		div32ap 9012	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2)))
37	31, 34, 35, 36	mp3an 1378	. . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2))
38		4d2e2 9444	. . . . . . 7 ⊢ (4 / 2) = 2
39	38	oveq1i 6085	. . . . . 6 ⊢ ((4 / 2) · π) = (2 · π)
40	30, 37, 39	3eqtr2i 2265	. . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π)
41	20, 40	breqtrdi 4166	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π))
42		pire 15810	. . . . . 6 ⊢ π ∈ ℝ
43	42	rexri 8373	. . . . 5 ⊢ π ∈ ℝ^*
44		2re 9353	. . . . . . 7 ⊢ 2 ∈ ℝ
45	44, 42	remulcli 8330	. . . . . 6 ⊢ (2 · π) ∈ ℝ
46	45	rexri 8373	. . . . 5 ⊢ (2 · π) ∈ ℝ^*
47		elioo2 10302	. . . . 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 430	. . . 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 1212	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π)))
50		sinq34lt0t 15855	. . 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 4149	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 1009 = wceq 1402 ∈ wcel 2209 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 ℂcc 8167 ℝcr 8168 0cc0 8169 1c1 8170 + caddc 8172 · cmul 8174 ℝ^*cxr 8349 < clt 8350 # cap 8899 / cdiv 8992 2c2 9334 3c3 9335 4c4 9336 (,)cioo 10269 sincsin 12389 cosccos 12390 πcpi 12392

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289 ax-pre-suploc 8290 ax-addf 8291 ax-mulf 8292

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-disj 4102 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-of 6292 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-oadd 6681 df-er 6797 df-map 6914 df-pm 6915 df-en 7013 df-dom 7014 df-fin 7015 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-xneg 10153 df-xadd 10154 df-ioo 10273 df-ioc 10274 df-ico 10275 df-icc 10276 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-fac 11142 df-bc 11164 df-ihash 11193 df-shft 11558 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-sumdc 12098 df-ef 12393 df-sin 12395 df-cos 12396 df-pi 12398 df-rest 13572 df-topgen 13591 df-psmet 14852 df-xmet 14853 df-met 14854 df-bl 14855 df-mopn 14856 df-top 15022 df-topon 15035 df-bases 15067 df-ntr 15120 df-cn 15212 df-cnp 15213 df-tx 15277 df-cncf 15595 df-limced 15680 df-dvap 15681

This theorem is referenced by: coseq0q4123 15858 cos02pilt1 15875

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