cosq14gt0 - Metamath Proof Explorer

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Theorem cosq14gt0 26751

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 26705	. . . . 5 ⊢ (π / 2) ∈ ℝ
2		elioore 13431	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℝ)
3		resubcl 11549	. . . . 5 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
4	1, 2, 3	sylancr 599	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ ℝ)
5		neghalfpirx 26707	. . . . . . 7 ⊢ -(π / 2) ∈ ℝ^*
6	1	rexri 11294	. . . . . . 7 ⊢ (π / 2) ∈ ℝ^*
7		elioo2 13442	. . . . . . 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 705	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ -(π / 2) < 𝐴 ∧ 𝐴 < (π / 2)))
9	8	simp3bi 1165	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 < (π / 2))
10		posdif 11734	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
11	2, 1, 10	sylancl 598	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (𝐴 < (π / 2) ↔ 0 < ((π / 2) − 𝐴)))
12	9, 11	mpbid 235	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < ((π / 2) − 𝐴))
13		picn 26697	. . . . . . . 8 ⊢ π ∈ ℂ
14		halfcl 12497	. . . . . . . 8 ⊢ (π ∈ ℂ → (π / 2) ∈ ℂ)
15	13, 14	ax-mp 5	. . . . . . 7 ⊢ (π / 2) ∈ ℂ
16	15	negcli 11553	. . . . . . 7 ⊢ -(π / 2) ∈ ℂ
17	13, 15	negsubi 11563	. . . . . . . 8 ⊢ (π + -(π / 2)) = (π − (π / 2))
18		pidiv2halves 26708	. . . . . . . . 9 ⊢ ((π / 2) + (π / 2)) = π
19	13, 15, 15, 18	subaddrii 11574	. . . . . . . 8 ⊢ (π − (π / 2)) = (π / 2)
20	17, 19	eqtri 2783	. . . . . . 7 ⊢ (π + -(π / 2)) = (π / 2)
21	15, 13, 16, 20	subaddrii 11574	. . . . . 6 ⊢ ((π / 2) − π) = -(π / 2)
22	8	simp2bi 1164	. . . . . 6 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → -(π / 2) < 𝐴)
23	21, 22	eqbrtrid 5140	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − π) < 𝐴)
24		pire 26695	. . . . . . 7 ⊢ π ∈ ℝ
25		ltsub23 11721	. . . . . . 7 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ π ∈ ℝ) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
26	1, 24, 25	mp3an13 1481	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
27	2, 26	syl 18	. . . . 5 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (((π / 2) − 𝐴) < π ↔ ((π / 2) − π) < 𝐴))
28	23, 27	mpbird 260	. . . 4 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) < π)
29		0xr 11283	. . . . 5 ⊢ 0 ∈ ℝ^*
30	24	rexri 11294	. . . . 5 ⊢ π ∈ ℝ^*
31		elioo2 13442	. . . . 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 705	. . . 4 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) ↔ (((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < π))
33	4, 12, 28, 32	syl3anbrc 1362	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → ((π / 2) − 𝐴) ∈ (0(,)π))
34		sinq12gt0 26748	. . 3 ⊢ (((π / 2) − 𝐴) ∈ (0(,)π) → 0 < (sin‘((π / 2) − 𝐴)))
35	33, 34	syl 18	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
36	2	recnd 11264	. . 3 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 𝐴 ∈ ℂ)
37		sinhalfpim 26734	. . 3 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
38	36, 37	syl 18	. 2 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
39	35, 38	breqtrd 5131	1 ⊢ (𝐴 ∈ (-(π / 2)(,)(π / 2)) → 0 < (cos‘𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 ‘cfv 6533 (class class class)co 7414 ℂcc 11125 ℝcr 11126 0cc0 11127 + caddc 11130 ℝ^*cxr 11269 < clt 11270 − cmin 11468 -cneg 11469 / cdiv 11898 2c2 12322 (,)cioo 13401 sincsin 16152 cosccos 16153 πcpi 16155

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-sup 9415 df-inf 9416 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ioo 13405 df-ioc 13406 df-ico 13407 df-icc 13408 df-fz 13565 df-fzo 13713 df-fl 13856 df-seq 14069 df-exp 14129 df-fac 14341 df-bc 14370 df-hash 14398 df-shft 15143 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-limsup 15561 df-clim 15578 df-rlim 15579 df-sum 15777 df-ef 16156 df-sin 16158 df-cos 16159 df-pi 16161 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-rest 17510 df-topn 17511 df-0g 17529 df-gsum 17530 df-topgen 17531 df-pt 17532 df-prds 17535 df-xrs 17591 df-qtop 17596 df-imas 17597 df-xps 17599 df-mre 17673 df-mrc 17674 df-acs 17676 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-submnd 18895 df-mulg 19194 df-cntz 19447 df-cmn 19912 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-fbas 21585 df-fg 21586 df-cnfld 21589 df-top 23122 df-topon 23139 df-topsp 23161 df-bases 23174 df-cld 23247 df-ntr 23248 df-cls 23249 df-nei 23326 df-lp 23364 df-perf 23365 df-cn 23455 df-cnp 23456 df-haus 23543 df-tx 23791 df-hmeo 23984 df-fil 24075 df-fm 24167 df-flim 24168 df-flf 24169 df-xms 24549 df-ms 24550 df-tms 24551 df-cncf 25109 df-limc 26096 df-dv 26097

This theorem is used by: tanord1 26777 logcnlem4 26885 asinsinlem 27131 goldrapos 47751

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