sincosq1sgn - Metamath Proof Explorer

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Theorem sincosq1sgn 25683

Description: The signs of the sine and cosine functions in the first quadrant. (Contributed by Paul Chapman, 24-Jan-2008.)

**Assertion**
Ref	Expression
sincosq1sgn	⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))

**Proof of Theorem sincosq1sgn**
Step	Hyp	Ref	Expression
1		0xr 11050	. . 3 ⊢ 0 ∈ ℝ^*
2		halfpire 25649	. . . 4 ⊢ (π / 2) ∈ ℝ
3	2	rexri 11061	. . 3 ⊢ (π / 2) ∈ ℝ^*
4		elioo2 13148	. . 3 ⊢ ((0 ∈ ℝ^* ∧ (π / 2) ∈ ℝ^*) → (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2))))
5	1, 3, 4	mp2an 688	. 2 ⊢ (𝐴 ∈ (0(,)(π / 2)) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)))
6		sincosq1lem 25682	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))
7		resubcl 11313	. . . . . . . 8 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((π / 2) − 𝐴) ∈ ℝ)
8	2, 7	mpan 686	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → ((π / 2) − 𝐴) ∈ ℝ)
9		sincosq1lem 25682	. . . . . . 7 ⊢ ((((π / 2) − 𝐴) ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
10	8, 9	syl3an1 1161	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴)))
11	10	3expib 1120	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) → 0 < (sin‘((π / 2) − 𝐴))))
12		0re 11005	. . . . . . . . 9 ⊢ 0 ∈ ℝ
13		ltsub13 11484	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
14	12, 2, 13	mp3an12 1449	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < ((π / 2) − 0)))
15	2	recni 11017	. . . . . . . . . 10 ⊢ (π / 2) ∈ ℂ
16	15	subid1i 11321	. . . . . . . . 9 ⊢ ((π / 2) − 0) = (π / 2)
17	16	breq2i 5085	. . . . . . . 8 ⊢ (𝐴 < ((π / 2) − 0) ↔ 𝐴 < (π / 2))
18	14, 17	bitrdi 286	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (0 < ((π / 2) − 𝐴) ↔ 𝐴 < (π / 2)))
19		ltsub23 11483	. . . . . . . . 9 ⊢ (((π / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
20	2, 2, 19	mp3an13 1450	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < (π / 2) ↔ ((π / 2) − (π / 2)) < 𝐴))
21	15	subidi 11320	. . . . . . . . 9 ⊢ ((π / 2) − (π / 2)) = 0
22	21	breq1i 5084	. . . . . . . 8 ⊢ (((π / 2) − (π / 2)) < 𝐴 ↔ 0 < 𝐴)
23	20, 22	bitrdi 286	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (((π / 2) − 𝐴) < (π / 2) ↔ 0 < 𝐴))
24	18, 23	anbi12d 630	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) ↔ (𝐴 < (π / 2) ∧ 0 < 𝐴)))
25	24	biancomd 463	. . . . 5 ⊢ (𝐴 ∈ ℝ → ((0 < ((π / 2) − 𝐴) ∧ ((π / 2) − 𝐴) < (π / 2)) ↔ (0 < 𝐴 ∧ 𝐴 < (π / 2))))
26		recn 10989	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
27		sinhalfpim 25678	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (sin‘((π / 2) − 𝐴)) = (cos‘𝐴))
29	28	breq2d 5089	. . . . 5 ⊢ (𝐴 ∈ ℝ → (0 < (sin‘((π / 2) − 𝐴)) ↔ 0 < (cos‘𝐴)))
30	11, 25, 29	3imtr3d 292	. . . 4 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (cos‘𝐴)))
31	30	3impib 1114	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (cos‘𝐴))
32	6, 31	jca 511	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))
33	5, 32	sylbi 216	1 ⊢ (𝐴 ∈ (0(,)(π / 2)) → (0 < (sin‘𝐴) ∧ 0 < (cos‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 class class class wbr 5077 ‘cfv 6447 (class class class)co 7295 ℂcc 10897 ℝcr 10898 0cc0 10899 ℝ^*cxr 11036 < clt 11037 − cmin 11233 / cdiv 11660 2c2 12056 (,)cioo 13107 sincsin 15801 cosccos 15802 πcpi 15804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 ax-inf2 9427 ax-cnex 10955 ax-resscn 10956 ax-1cn 10957 ax-icn 10958 ax-addcl 10959 ax-addrcl 10960 ax-mulcl 10961 ax-mulrcl 10962 ax-mulcom 10963 ax-addass 10964 ax-mulass 10965 ax-distr 10966 ax-i2m1 10967 ax-1ne0 10968 ax-1rid 10969 ax-rnegex 10970 ax-rrecex 10971 ax-cnre 10972 ax-pre-lttri 10973 ax-pre-lttrn 10974 ax-pre-ltadd 10975 ax-pre-mulgt0 10976 ax-pre-sup 10977 ax-addf 10978 ax-mulf 10979

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-iin 4930 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-isom 6456 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-of 7553 df-om 7733 df-1st 7851 df-2nd 7852 df-supp 7998 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-rdg 8261 df-1o 8317 df-2o 8318 df-er 8518 df-map 8637 df-pm 8638 df-ixp 8706 df-en 8754 df-dom 8755 df-sdom 8756 df-fin 8757 df-fsupp 9157 df-fi 9198 df-sup 9229 df-inf 9230 df-oi 9297 df-card 9725 df-pnf 11039 df-mnf 11040 df-xr 11041 df-ltxr 11042 df-le 11043 df-sub 11235 df-neg 11236 df-div 11661 df-nn 12002 df-2 12064 df-3 12065 df-4 12066 df-5 12067 df-6 12068 df-7 12069 df-8 12070 df-9 12071 df-n0 12262 df-z 12348 df-dec 12466 df-uz 12611 df-q 12717 df-rp 12759 df-xneg 12876 df-xadd 12877 df-xmul 12878 df-ioo 13111 df-ioc 13112 df-ico 13113 df-icc 13114 df-fz 13268 df-fzo 13411 df-fl 13540 df-seq 13750 df-exp 13811 df-fac 14016 df-bc 14045 df-hash 14073 df-shft 14806 df-cj 14838 df-re 14839 df-im 14840 df-sqrt 14974 df-abs 14975 df-limsup 15208 df-clim 15225 df-rlim 15226 df-sum 15426 df-ef 15805 df-sin 15807 df-cos 15808 df-pi 15810 df-struct 16876 df-sets 16893 df-slot 16911 df-ndx 16923 df-base 16941 df-ress 16970 df-plusg 17003 df-mulr 17004 df-starv 17005 df-sca 17006 df-vsca 17007 df-ip 17008 df-tset 17009 df-ple 17010 df-ds 17012 df-unif 17013 df-hom 17014 df-cco 17015 df-rest 17161 df-topn 17162 df-0g 17180 df-gsum 17181 df-topgen 17182 df-pt 17183 df-prds 17186 df-xrs 17241 df-qtop 17246 df-imas 17247 df-xps 17249 df-mre 17323 df-mrc 17324 df-acs 17326 df-mgm 18354 df-sgrp 18403 df-mnd 18414 df-submnd 18459 df-mulg 18729 df-cntz 18951 df-cmn 19416 df-psmet 20617 df-xmet 20618 df-met 20619 df-bl 20620 df-mopn 20621 df-fbas 20622 df-fg 20623 df-cnfld 20626 df-top 22071 df-topon 22088 df-topsp 22110 df-bases 22124 df-cld 22198 df-ntr 22199 df-cls 22200 df-nei 22277 df-lp 22315 df-perf 22316 df-cn 22406 df-cnp 22407 df-haus 22494 df-tx 22741 df-hmeo 22934 df-fil 23025 df-fm 23117 df-flim 23118 df-flf 23119 df-xms 23501 df-ms 23502 df-tms 23503 df-cncf 24069 df-limc 25058 df-dv 25059

This theorem is referenced by: sincosq2sgn 25684 coseq00topi 25687 tanrpcl 25689 tangtx 25690 tanabsge 25691 sincos6thpi 25700 tanord1 25721 basellem3 26260 basellem4 26261 basellem8 26265

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