cosq14gt0 - Intuitionistic Logic Explorer

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Theorem cosq14gt0 13393

Description: The cosine of a number strictly between -π / 2 and π / 2 is positive. (Contributed by Mario Carneiro, 25-Feb-2015.)

**Assertion**
Ref	Expression
cosq14gt0	⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴))

**Proof of Theorem cosq14gt0**
Step	Hyp	Ref	Expression
1		halfpire 13353	. . . . 5 ⊢ (π / 2) ∈ ℝ
2		elioore 9848	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
3		resubcl 8162	. . . . 5 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
4	1, 2, 3	sylancr 411	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ)
5		neghalfpirx 13355	. . . . . . 7 ⊢ -(π / 2) ∈ ℝ^*
6	1	rexri 7956	. . . . . . 7 ⊢ (π / 2) ∈ ℝ^*
7		elioo2 9857	. . . . . . 7 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2))))
8	5, 6, 7	mp2an 423	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
9	8	simp3bi 1004	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
10		posdif 8353	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
11	2, 1, 10	sylancl 410	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
12	9, 11	mpbid 146	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < ((π / 2) − 𝐴))
13		picn 13348	. . . . . . . 8 ⊢ π ∈ ℂ
14		halfcl 9083	. . . . . . . 8 ⊢ (π ∈ ℂ → (π / 2) ∈ ℂ)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℂ
16	15	negcli 8166	. . . . . . 7 ⊢ -(π / 2) ∈ ℂ
17	13, 15	negsubi 8176	. . . . . . . 8 ⊢ (π + -(π / 2)) = (π − (π / 2))
18		pidiv2halves 13356	. . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π
19	13, 15, 15, 18	subaddrii 8187	. . . . . . . 8 ⊢ (π − (π / 2)) = (π / 2)
20	17, 19	eqtri 2186	. . . . . . 7 ⊢ (π + -(π / 2)) = (π / 2)
21	15, 13, 16, 20	subaddrii 8187	. . . . . 6 ⊢ ((π / 2) − π) = -(π / 2)
22	8	simp2bi 1003	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
23	21, 22	eqbrtrid 4017	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
24		pire 13347	. . . . . . 7 ⊢ π ∈ ℝ
25		ltsub23 8340	. . . . . . 7 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ π ∈ ℝ) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
26	1, 24, 25	mp3an13 1318	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
27	2, 26	syl 14	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
28	23, 27	mpbird 166	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
29		0xr 7945	. . . . 5 ⊢ 0 ∈ ℝ^*
30	24	rexri 7956	. . . . 5 ⊢ π ∈ ℝ^*
31		elioo2 9857	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^*) → (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π)))
32	29, 30, 31	mp2an 423	. . . 4 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π))
33	4, 12, 28, 32	syl3anbrc 1171	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ (0(,)π))
34		sinq12gt0 13391	. . 3 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) → 0 < (sin‘((π / 2) − 𝐴)))
35	33, 34	syl 14	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
36	2	recnd 7927	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
37		sinhalfpim 13382	. . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
38	36, 37	syl 14	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
39	35, 38	breqtrd 4008	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 ℂcc 7751 ℝcr 7752 0cc0 7753 + caddc 7756 ℝ^*cxr 7932 < clt 7933 − cmin 8069 -cneg 8070 / cdiv 8568 2c2 8908 (,)cioo 9824 sincsin 11585 cosccos 11586 πcpi 11588

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 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873 ax-pre-suploc 7874 ax-addf 7875 ax-mulf 7876

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 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-disj 3960 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-isom 5197 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-of 6050 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-frec 6359 df-1o 6384 df-oadd 6388 df-er 6501 df-map 6616 df-pm 6617 df-en 6707 df-dom 6708 df-fin 6709 df-sup 6949 df-inf 6950 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-5 8919 df-6 8920 df-7 8921 df-8 8922 df-9 8923 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-xneg 9708 df-xadd 9709 df-ioo 9828 df-ioc 9829 df-ico 9830 df-icc 9831 df-fz 9945 df-fzo 10078 df-seqfrec 10381 df-exp 10455 df-fac 10639 df-bc 10661 df-ihash 10689 df-shft 10757 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-clim 11220 df-sumdc 11295 df-ef 11589 df-sin 11591 df-cos 11592 df-pi 11594 df-rest 12558 df-topgen 12577 df-psmet 12627 df-xmet 12628 df-met 12629 df-bl 12630 df-mopn 12631 df-top 12636 df-topon 12649 df-bases 12681 df-ntr 12736 df-cn 12828 df-cnp 12829 df-tx 12893 df-cncf 13198 df-limced 13265 df-dvap 13266

This theorem is referenced by: coseq0q4123 13395

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