cosq14gt0 - Intuitionistic Logic Explorer

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Theorem cosq14gt0 15389

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 15349	. . . . 5 ⊢ (π / 2) ∈ ℝ
2		elioore 10064	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
3		resubcl 8366	. . . . 5 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
4	1, 2, 3	sylancr 414	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ)
5		neghalfpirx 15351	. . . . . . 7 ⊢ -(π / 2) ∈ ℝ^*
6	1	rexri 8160	. . . . . . 7 ⊢ (π / 2) ∈ ℝ^*
7		elioo2 10073	. . . . . . 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 426	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
9	8	simp3bi 1017	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
10		posdif 8558	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
11	2, 1, 10	sylancl 413	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
12	9, 11	mpbid 147	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < ((π / 2) − 𝐴))
13		picn 15344	. . . . . . . 8 ⊢ π ∈ ℂ
14		halfcl 9293	. . . . . . . 8 ⊢ (π ∈ ℂ → (π / 2) ∈ ℂ)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℂ
16	15	negcli 8370	. . . . . . 7 ⊢ -(π / 2) ∈ ℂ
17	13, 15	negsubi 8380	. . . . . . . 8 ⊢ (π + -(π / 2)) = (π − (π / 2))
18		pidiv2halves 15352	. . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π
19	13, 15, 15, 18	subaddrii 8391	. . . . . . . 8 ⊢ (π − (π / 2)) = (π / 2)
20	17, 19	eqtri 2227	. . . . . . 7 ⊢ (π + -(π / 2)) = (π / 2)
21	15, 13, 16, 20	subaddrii 8391	. . . . . 6 ⊢ ((π / 2) − π) = -(π / 2)
22	8	simp2bi 1016	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
23	21, 22	eqbrtrid 4089	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
24		pire 15343	. . . . . . 7 ⊢ π ∈ ℝ
25		ltsub23 8545	. . . . . . 7 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ π ∈ ℝ) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
26	1, 24, 25	mp3an13 1341	. . . . . 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 167	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
29		0xr 8149	. . . . 5 ⊢ 0 ∈ ℝ^*
30	24	rexri 8160	. . . . 5 ⊢ π ∈ ℝ^*
31		elioo2 10073	. . . . 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 426	. . . 4 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π))
33	4, 12, 28, 32	syl3anbrc 1184	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ (0(,)π))
34		sinq12gt0 15387	. . 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 8131	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
37		sinhalfpim 15378	. . 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 4080	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 981 = wceq 1373 ∈ wcel 2177 class class class wbr 4054 ‘cfv 5285 (class class class)co 5962 ℂcc 7953 ℝcr 7954 0cc0 7955 + caddc 7958 ℝ^*cxr 8136 < clt 8137 − cmin 8273 -cneg 8274 / cdiv 8775 2c2 9117 (,)cioo 10040 sincsin 12040 cosccos 12041 πcpi 12043

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4170 ax-sep 4173 ax-nul 4181 ax-pow 4229 ax-pr 4264 ax-un 4493 ax-setind 4598 ax-iinf 4649 ax-cnex 8046 ax-resscn 8047 ax-1cn 8048 ax-1re 8049 ax-icn 8050 ax-addcl 8051 ax-addrcl 8052 ax-mulcl 8053 ax-mulrcl 8054 ax-addcom 8055 ax-mulcom 8056 ax-addass 8057 ax-mulass 8058 ax-distr 8059 ax-i2m1 8060 ax-0lt1 8061 ax-1rid 8062 ax-0id 8063 ax-rnegex 8064 ax-precex 8065 ax-cnre 8066 ax-pre-ltirr 8067 ax-pre-ltwlin 8068 ax-pre-lttrn 8069 ax-pre-apti 8070 ax-pre-ltadd 8071 ax-pre-mulgt0 8072 ax-pre-mulext 8073 ax-arch 8074 ax-caucvg 8075 ax-pre-suploc 8076 ax-addf 8077 ax-mulf 8078

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3860 df-int 3895 df-iun 3938 df-disj 4031 df-br 4055 df-opab 4117 df-mpt 4118 df-tr 4154 df-id 4353 df-po 4356 df-iso 4357 df-iord 4426 df-on 4428 df-ilim 4429 df-suc 4431 df-iom 4652 df-xp 4694 df-rel 4695 df-cnv 4696 df-co 4697 df-dm 4698 df-rn 4699 df-res 4700 df-ima 4701 df-iota 5246 df-fun 5287 df-fn 5288 df-f 5289 df-f1 5290 df-fo 5291 df-f1o 5292 df-fv 5293 df-isom 5294 df-riota 5917 df-ov 5965 df-oprab 5966 df-mpo 5967 df-of 6176 df-1st 6244 df-2nd 6245 df-recs 6409 df-irdg 6474 df-frec 6495 df-1o 6520 df-oadd 6524 df-er 6638 df-map 6755 df-pm 6756 df-en 6846 df-dom 6847 df-fin 6848 df-sup 7107 df-inf 7108 df-pnf 8139 df-mnf 8140 df-xr 8141 df-ltxr 8142 df-le 8143 df-sub 8275 df-neg 8276 df-reap 8678 df-ap 8685 df-div 8776 df-inn 9067 df-2 9125 df-3 9126 df-4 9127 df-5 9128 df-6 9129 df-7 9130 df-8 9131 df-9 9132 df-n0 9326 df-z 9403 df-uz 9679 df-q 9771 df-rp 9806 df-xneg 9924 df-xadd 9925 df-ioo 10044 df-ioc 10045 df-ico 10046 df-icc 10047 df-fz 10161 df-fzo 10295 df-seqfrec 10625 df-exp 10716 df-fac 10903 df-bc 10925 df-ihash 10953 df-shft 11211 df-cj 11238 df-re 11239 df-im 11240 df-rsqrt 11394 df-abs 11395 df-clim 11675 df-sumdc 11750 df-ef 12044 df-sin 12046 df-cos 12047 df-pi 12049 df-rest 13158 df-topgen 13177 df-psmet 14390 df-xmet 14391 df-met 14392 df-bl 14393 df-mopn 14394 df-top 14555 df-topon 14568 df-bases 14600 df-ntr 14653 df-cn 14745 df-cnp 14746 df-tx 14810 df-cncf 15128 df-limced 15213 df-dvap 15214

This theorem is referenced by: coseq0q4123 15391

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