sinq12ge0 - Metamath Proof Explorer

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Theorem sinq12ge0 25748

Description: The sine of a number between 0 and π is nonnegative. (Contributed by Mario Carneiro, 13-May-2014.)

**Assertion**
Ref	Expression
sinq12ge0	⊢ (𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴))

**Proof of Theorem sinq12ge0**
Step	Hyp	Ref	Expression
1		0re 11057	. . . . . . . 8 ⊢ 0 ∈ ℝ
2		pire 25698	. . . . . . . 8 ⊢ π ∈ ℝ
3	1, 2	elicc2i 13225	. . . . . . 7 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1144	. . . . . 6 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		rexr 11101	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → 0 ∈ ℝ^*)
6		rexr 11101	. . . . . . . . . 10 ⊢ (π ∈ ℝ → π ∈ ℝ^*)
7		elioo2 13200	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^*) → (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π)))
8	5, 6, 7	syl2an 596	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π)))
9	1, 2, 8	mp2an 689	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π))
10		sinq12gt0 25747	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴))
11	9, 10	sylbir 234	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘𝐴))
12	11	3expib 1121	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘𝐴)))
13	4, 12	syl 17	. . . . 5 ⊢ (𝐴 ∈ (0[,]π) → ((0 < 𝐴 ∧ 𝐴 < π) → 0 < (sin‘𝐴)))
14	4	resincld 15931	. . . . . 6 ⊢ (𝐴 ∈ (0[,]π) → (sin‘𝐴) ∈ ℝ)
15		ltle 11143	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ (sin‘𝐴) ∈ ℝ) → (0 < (sin‘𝐴) → 0 ≤ (sin‘𝐴)))
16	1, 14, 15	sylancr 587	. . . . 5 ⊢ (𝐴 ∈ (0[,]π) → (0 < (sin‘𝐴) → 0 ≤ (sin‘𝐴)))
17	13, 16	syld 47	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → ((0 < 𝐴 ∧ 𝐴 < π) → 0 ≤ (sin‘𝐴)))
18	17	expd 416	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (0 < 𝐴 → (𝐴 < π → 0 ≤ (sin‘𝐴))))
19		0le0 12154	. . . . . 6 ⊢ 0 ≤ 0
20		sin0 15937	. . . . . 6 ⊢ (sin‘0) = 0
21	19, 20	breqtrri 5114	. . . . 5 ⊢ 0 ≤ (sin‘0)
22		fveq2 6812	. . . . 5 ⊢ (0 = 𝐴 → (sin‘0) = (sin‘𝐴))
23	21, 22	breqtrid 5124	. . . 4 ⊢ (0 = 𝐴 → 0 ≤ (sin‘𝐴))
24	23	a1i13 27	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (0 = 𝐴 → (𝐴 < π → 0 ≤ (sin‘𝐴))))
25	3	simp2bi 1145	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ 𝐴)
26		leloe 11141	. . . . 5 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
27	1, 4, 26	sylancr 587	. . . 4 ⊢ (𝐴 ∈ (0[,]π) → (0 ≤ 𝐴 ↔ (0 < 𝐴 ∨ 0 = 𝐴)))
28	25, 27	mpbid 231	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (0 < 𝐴 ∨ 0 = 𝐴))
29	18, 24, 28	mpjaod 857	. 2 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 < π → 0 ≤ (sin‘𝐴)))
30		sinpi 25697	. . . . 5 ⊢ (sin‘π) = 0
31	19, 30	breqtrri 5114	. . . 4 ⊢ 0 ≤ (sin‘π)
32		fveq2 6812	. . . 4 ⊢ (𝐴 = π → (sin‘𝐴) = (sin‘π))
33	31, 32	breqtrrid 5125	. . 3 ⊢ (𝐴 = π → 0 ≤ (sin‘𝐴))
34	33	a1i 11	. 2 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 = π → 0 ≤ (sin‘𝐴)))
35	3	simp3bi 1146	. . 3 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ≤ π)
36		leloe 11141	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ π ∈ ℝ) → (𝐴 ≤ π ↔ (𝐴 < π ∨ 𝐴 = π)))
37	4, 2, 36	sylancl 586	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 ≤ π ↔ (𝐴 < π ∨ 𝐴 = π)))
38	35, 37	mpbid 231	. 2 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 < π ∨ 𝐴 = π))
39	29, 34, 38	mpjaod 857	1 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 class class class wbr 5087 ‘cfv 6466 (class class class)co 7317 ℝcr 10950 0cc0 10951 ℝ^*cxr 11088 < clt 11089 ≤ cle 11090 (,)cioo 13159 [,]cicc 13162 sincsin 15852 πcpi 15855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630 ax-inf2 9477 ax-cnex 11007 ax-resscn 11008 ax-1cn 11009 ax-icn 11010 ax-addcl 11011 ax-addrcl 11012 ax-mulcl 11013 ax-mulrcl 11014 ax-mulcom 11015 ax-addass 11016 ax-mulass 11017 ax-distr 11018 ax-i2m1 11019 ax-1ne0 11020 ax-1rid 11021 ax-rnegex 11022 ax-rrecex 11023 ax-cnre 11024 ax-pre-lttri 11025 ax-pre-lttrn 11026 ax-pre-ltadd 11027 ax-pre-mulgt0 11028 ax-pre-sup 11029 ax-addf 11030 ax-mulf 11031

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4851 df-int 4893 df-iun 4939 df-iin 4940 df-br 5088 df-opab 5150 df-mpt 5171 df-tr 5205 df-id 5507 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5563 df-se 5564 df-we 5565 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-pred 6225 df-ord 6292 df-on 6293 df-lim 6294 df-suc 6295 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-isom 6475 df-riota 7274 df-ov 7320 df-oprab 7321 df-mpo 7322 df-of 7575 df-om 7760 df-1st 7878 df-2nd 7879 df-supp 8027 df-frecs 8146 df-wrecs 8177 df-recs 8251 df-rdg 8290 df-1o 8346 df-2o 8347 df-er 8548 df-map 8667 df-pm 8668 df-ixp 8736 df-en 8784 df-dom 8785 df-sdom 8786 df-fin 8787 df-fsupp 9206 df-fi 9247 df-sup 9278 df-inf 9279 df-oi 9346 df-card 9775 df-pnf 11091 df-mnf 11092 df-xr 11093 df-ltxr 11094 df-le 11095 df-sub 11287 df-neg 11288 df-div 11713 df-nn 12054 df-2 12116 df-3 12117 df-4 12118 df-5 12119 df-6 12120 df-7 12121 df-8 12122 df-9 12123 df-n0 12314 df-z 12400 df-dec 12518 df-uz 12663 df-q 12769 df-rp 12811 df-xneg 12928 df-xadd 12929 df-xmul 12930 df-ioo 13163 df-ioc 13164 df-ico 13165 df-icc 13166 df-fz 13320 df-fzo 13463 df-fl 13592 df-seq 13802 df-exp 13863 df-fac 14068 df-bc 14097 df-hash 14125 df-shft 14857 df-cj 14889 df-re 14890 df-im 14891 df-sqrt 15025 df-abs 15026 df-limsup 15259 df-clim 15276 df-rlim 15277 df-sum 15477 df-ef 15856 df-sin 15858 df-cos 15859 df-pi 15861 df-struct 16925 df-sets 16942 df-slot 16960 df-ndx 16972 df-base 16990 df-ress 17019 df-plusg 17052 df-mulr 17053 df-starv 17054 df-sca 17055 df-vsca 17056 df-ip 17057 df-tset 17058 df-ple 17059 df-ds 17061 df-unif 17062 df-hom 17063 df-cco 17064 df-rest 17210 df-topn 17211 df-0g 17229 df-gsum 17230 df-topgen 17231 df-pt 17232 df-prds 17235 df-xrs 17290 df-qtop 17295 df-imas 17296 df-xps 17298 df-mre 17372 df-mrc 17373 df-acs 17375 df-mgm 18403 df-sgrp 18452 df-mnd 18463 df-submnd 18508 df-mulg 18777 df-cntz 18999 df-cmn 19463 df-psmet 20672 df-xmet 20673 df-met 20674 df-bl 20675 df-mopn 20676 df-fbas 20677 df-fg 20678 df-cnfld 20681 df-top 22126 df-topon 22143 df-topsp 22165 df-bases 22179 df-cld 22253 df-ntr 22254 df-cls 22255 df-nei 22332 df-lp 22370 df-perf 22371 df-cn 22461 df-cnp 22462 df-haus 22549 df-tx 22796 df-hmeo 22989 df-fil 23080 df-fm 23172 df-flim 23173 df-flf 23174 df-xms 23556 df-ms 23557 df-tms 23558 df-cncf 24124 df-limc 25113 df-dv 25114

This theorem is referenced by: cosq14ge0 25751 argimgt0 25850 sin2h 35839

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