sinq12ge0 - Metamath Proof Explorer

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Theorem sinq12ge0 26564

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 11260	. . . . . . . 8 ⊢ 0 ∈ ℝ
2		pire 26514	. . . . . . . 8 ⊢ π ∈ ℝ
3	1, 2	elicc2i 13449	. . . . . . 7 ⊢ (𝐴 ∈ (0[,]π) ↔ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐴 ≤ π))
4	3	simp1bi 1144	. . . . . 6 ⊢ (𝐴 ∈ (0[,]π) → 𝐴 ∈ ℝ)
5		rexr 11304	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → 0 ∈ ℝ^*)
6		rexr 11304	. . . . . . . . . 10 ⊢ (π ∈ ℝ → π ∈ ℝ^*)
7		elioo2 13424	. . . . . . . . . 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 692	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)π) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < π))
10		sinq12gt0 26563	. . . . . . . 8 ⊢ (𝐴 ∈ (0(,)π) → 0 < (sin‘𝐴))
11	9, 10	sylbir 235	. . . . . . 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 16175	. . . . . 6 ⊢ (𝐴 ∈ (0[,]π) → (sin‘𝐴) ∈ ℝ)
15		ltle 11346	. . . . . 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 415	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (0 < 𝐴 → (𝐴 < π → 0 ≤ (sin‘𝐴))))
19		0le0 12364	. . . . . 6 ⊢ 0 ≤ 0
20		sin0 16181	. . . . . 6 ⊢ (sin‘0) = 0
21	19, 20	breqtrri 5174	. . . . 5 ⊢ 0 ≤ (sin‘0)
22		fveq2 6906	. . . . 5 ⊢ (0 = 𝐴 → (sin‘0) = (sin‘𝐴))
23	21, 22	breqtrid 5184	. . . 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 11344	. . . . 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 232	. . 3 ⊢ (𝐴 ∈ (0[,]π) → (0 < 𝐴 ∨ 0 = 𝐴))
29	18, 24, 28	mpjaod 860	. 2 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 < π → 0 ≤ (sin‘𝐴)))
30		sinpi 26513	. . . . 5 ⊢ (sin‘π) = 0
31	19, 30	breqtrri 5174	. . . 4 ⊢ 0 ≤ (sin‘π)
32		fveq2 6906	. . . 4 ⊢ (𝐴 = π → (sin‘𝐴) = (sin‘π))
33	31, 32	breqtrrid 5185	. . 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 11344	. . . 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 232	. 2 ⊢ (𝐴 ∈ (0[,]π) → (𝐴 < π ∨ 𝐴 = π))
39	29, 34, 38	mpjaod 860	1 ⊢ (𝐴 ∈ (0[,]π) → 0 ≤ (sin‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 (,)cioo 13383 [,]cicc 13386 sincsin 16095 πcpi 16098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-xms 24345 df-ms 24346 df-tms 24347 df-cncf 24917 df-limc 25915 df-dv 25916

This theorem is referenced by: cosq14ge0 26567 argimgt0 26668 sin2h 37596

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