cosq23lt0 - Intuitionistic Logic Explorer

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Theorem cosq23lt0 13354

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 9844	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℝ)
2	1	recnd 7923	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 ∈ ℂ)
3		sinhalfpip 13341	. . 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 13313	. . . . . 6 ⊢ (π / 2) ∈ ℝ
6	5	a1i 9	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) ∈ ℝ)
7	6, 1	readdcld 7924	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ ℝ)
8		pidiv2halves 13316	. . . . 5 ⊢ ((π / 2) + (π / 2)) = π
9	5	rexri 7952	. . . . . . . 8 ⊢ (π / 2) ∈ ℝ^*
10		3re 8927	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
11	10, 5	remulcli 7909	. . . . . . . . 9 ⊢ (3 · (π / 2)) ∈ ℝ
12	11	rexri 7952	. . . . . . . 8 ⊢ (3 · (π / 2)) ∈ ℝ^*
13		elioo2 9853	. . . . . . . 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 423	. . . . . . 7 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) ↔ (𝐴 ∈ ℝ ∧ (π / 2) < 𝐴 ∧ 𝐴 < (3 · (π / 2))))
15	14	simp2bi 1003	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (π / 2) < 𝐴)
16	6, 1, 6, 15	ltadd2dd 8316	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + (π / 2)) < ((π / 2) + 𝐴))
17	8, 16	eqbrtrrid 4017	. . . 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 1004	. . . . . 6 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → 𝐴 < (3 · (π / 2)))
20	1, 18, 6, 19	ltadd2dd 8316	. . . . 5 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < ((π / 2) + (3 · (π / 2))))
21		ax-1cn 7842	. . . . . . . 8 ⊢ 1 ∈ ℂ
22		3cn 8928	. . . . . . . 8 ⊢ 3 ∈ ℂ
23	5	recni 7907	. . . . . . . 8 ⊢ (π / 2) ∈ ℂ
24	21, 22, 23	adddiri 7906	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = ((1 · (π / 2)) + (3 · (π / 2)))
25		3p1e4 8988	. . . . . . . . 9 ⊢ (3 + 1) = 4
26	22, 21, 25	addcomli 8039	. . . . . . . 8 ⊢ (1 + 3) = 4
27	26	oveq1i 5851	. . . . . . 7 ⊢ ((1 + 3) · (π / 2)) = (4 · (π / 2))
28	23	mulid2i 7898	. . . . . . . 8 ⊢ (1 · (π / 2)) = (π / 2)
29	28	oveq1i 5851	. . . . . . 7 ⊢ ((1 · (π / 2)) + (3 · (π / 2))) = ((π / 2) + (3 · (π / 2)))
30	24, 27, 29	3eqtr3ri 2195	. . . . . 6 ⊢ ((π / 2) + (3 · (π / 2))) = (4 · (π / 2))
31		4cn 8931	. . . . . . 7 ⊢ 4 ∈ ℂ
32		2cn 8924	. . . . . . . 8 ⊢ 2 ∈ ℂ
33		2ap0 8946	. . . . . . . 8 ⊢ 2 # 0
34	32, 33	pm3.2i 270	. . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 # 0)
35		picn 13308	. . . . . . 7 ⊢ π ∈ ℂ
36		div32ap 8584	. . . . . . 7 ⊢ ((4 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 # 0) ∧ π ∈ ℂ) → ((4 / 2) · π) = (4 · (π / 2)))
37	31, 34, 35, 36	mp3an 1327	. . . . . 6 ⊢ ((4 / 2) · π) = (4 · (π / 2))
38		4d2e2 9013	. . . . . . 7 ⊢ (4 / 2) = 2
39	38	oveq1i 5851	. . . . . 6 ⊢ ((4 / 2) · π) = (2 · π)
40	30, 37, 39	3eqtr2i 2192	. . . . 5 ⊢ ((π / 2) + (3 · (π / 2))) = (2 · π)
41	20, 40	breqtrdi 4022	. . . 4 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) < (2 · π))
42		pire 13307	. . . . . 6 ⊢ π ∈ ℝ
43	42	rexri 7952	. . . . 5 ⊢ π ∈ ℝ^*
44		2re 8923	. . . . . . 7 ⊢ 2 ∈ ℝ
45	44, 42	remulcli 7909	. . . . . 6 ⊢ (2 · π) ∈ ℝ
46	45	rexri 7952	. . . . 5 ⊢ (2 · π) ∈ ℝ^*
47		elioo2 9853	. . . . 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 423	. . . 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 1171	. . 3 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → ((π / 2) + 𝐴) ∈ (π(,)(2 · π)))
50		sinq34lt0t 13352	. . 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 4005	1 ⊢ (𝐴 ∈ ((π / 2)(,)(3 · (π / 2))) → (cos‘𝐴) < 0)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 class class class wbr 3981 ‘cfv 5187 (class class class)co 5841 ℂcc 7747 ℝcr 7748 0cc0 7749 1c1 7750 + caddc 7752 · cmul 7754 ℝ^*cxr 7928 < clt 7929 # cap 8475 / cdiv 8564 2c2 8904 3c3 8905 4c4 8906 (,)cioo 9820 sincsin 11581 cosccos 11582 πcpi 11584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867 ax-arch 7868 ax-caucvg 7869 ax-pre-suploc 7870 ax-addf 7871 ax-mulf 7872

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-disj 3959 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-isom 5196 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-of 6049 df-1st 6105 df-2nd 6106 df-recs 6269 df-irdg 6334 df-frec 6355 df-1o 6380 df-oadd 6384 df-er 6497 df-map 6612 df-pm 6613 df-en 6703 df-dom 6704 df-fin 6705 df-sup 6945 df-inf 6946 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-5 8915 df-6 8916 df-7 8917 df-8 8918 df-9 8919 df-n0 9111 df-z 9188 df-uz 9463 df-q 9554 df-rp 9586 df-xneg 9704 df-xadd 9705 df-ioo 9824 df-ioc 9825 df-ico 9826 df-icc 9827 df-fz 9941 df-fzo 10074 df-seqfrec 10377 df-exp 10451 df-fac 10635 df-bc 10657 df-ihash 10685 df-shft 10753 df-cj 10780 df-re 10781 df-im 10782 df-rsqrt 10936 df-abs 10937 df-clim 11216 df-sumdc 11291 df-ef 11585 df-sin 11587 df-cos 11588 df-pi 11590 df-rest 12553 df-topgen 12572 df-psmet 12587 df-xmet 12588 df-met 12589 df-bl 12590 df-mopn 12591 df-top 12596 df-topon 12609 df-bases 12641 df-ntr 12696 df-cn 12788 df-cnp 12789 df-tx 12853 df-cncf 13158 df-limced 13225 df-dvap 13226

This theorem is referenced by: coseq0q4123 13355 cos02pilt1 13372

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