sincosq1lem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sincosq1lem		Structured version Visualization version GIF version

Theorem sincosq1lem 26439

Description: Lemma for sincosq1sgn 26440. (Contributed by Paul Chapman, 24-Jan-2008.)

**Assertion**
Ref	Expression
sincosq1lem	⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))

**Proof of Theorem sincosq1lem**
Step	Hyp	Ref	Expression
1		halfpire 26406	. . . . . 6 ⊢ (π / 2) ∈ ℝ
2		ltle 11238	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ) → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2)))
3	1, 2	mpan2 691	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < (π / 2) → 𝐴 ≤ (π / 2)))
4		pire 26399	. . . . . . . 8 ⊢ π ∈ ℝ
5		4re 12246	. . . . . . . 8 ⊢ 4 ∈ ℝ
6		pigt2lt4 26397	. . . . . . . . 9 ⊢ (2 < π ∧ π < 4)
7	6	simpri 485	. . . . . . . 8 ⊢ π < 4
8	4, 5, 7	ltleii 11273	. . . . . . 7 ⊢ π ≤ 4
9		2re 12236	. . . . . . . . 9 ⊢ 2 ∈ ℝ
10		2pos 12265	. . . . . . . . . 10 ⊢ 0 < 2
11	9, 10	pm3.2i 470	. . . . . . . . 9 ⊢ (2 ∈ ℝ ∧ 0 < 2)
12		ledivmul 12035	. . . . . . . . 9 ⊢ ((π ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((π / 2) ≤ 2 ↔ π ≤ (2 · 2)))
13	4, 9, 11, 12	mp3an 1463	. . . . . . . 8 ⊢ ((π / 2) ≤ 2 ↔ π ≤ (2 · 2))
14		2t2e4 12321	. . . . . . . . 9 ⊢ (2 · 2) = 4
15	14	breq2i 5110	. . . . . . . 8 ⊢ (π ≤ (2 · 2) ↔ π ≤ 4)
16	13, 15	bitri 275	. . . . . . 7 ⊢ ((π / 2) ≤ 2 ↔ π ≤ 4)
17	8, 16	mpbir 231	. . . . . 6 ⊢ (π / 2) ≤ 2
18		letr 11244	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ 2 ∈ ℝ) → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2))
19	1, 9, 18	mp3an23 1455	. . . . . 6 ⊢ (𝐴 ∈ ℝ → ((𝐴 ≤ (π / 2) ∧ (π / 2) ≤ 2) → 𝐴 ≤ 2))
20	17, 19	mpan2i 697	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ (π / 2) → 𝐴 ≤ 2))
21	3, 20	syld 47	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 < (π / 2) → 𝐴 ≤ 2))
22	21	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → (𝐴 < (π / 2) → 𝐴 ≤ 2))
23	22	3impia 1117	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 𝐴 ≤ 2)
24		0xr 11197	. . . 4 ⊢ 0 ∈ ℝ^*
25		elioc2 13346	. . . 4 ⊢ ((0 ∈ ℝ^* ∧ 2 ∈ ℝ) → (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2)))
26	24, 9, 25	mp2an 692	. . 3 ⊢ (𝐴 ∈ (0(,]2) ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2))
27		sin02gt0 16136	. . 3 ⊢ (𝐴 ∈ (0(,]2) → 0 < (sin‘𝐴))
28	26, 27	sylbir 235	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 ≤ 2) → 0 < (sin‘𝐴))
29	23, 28	syld3an3 1411	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 < 𝐴 ∧ 𝐴 < (π / 2)) → 0 < (sin‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ℝcr 11043 0cc0 11044 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 / cdiv 11811 2c2 12217 4c4 12219 (,]cioc 13283 sincsin 16005 πcpi 16008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: sincosq1sgn 26440 sinq12gt0 26449

W3C validator