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Mirrors > Home > MPE Home > Th. List > sinltx | Structured version Visualization version GIF version |
Description: The sine of a positive real number is less than its argument. (Contributed by Mario Carneiro, 29-Jul-2014.) |
Ref | Expression |
---|---|
sinltx | β’ (π΄ β β+ β (sinβπ΄) < π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12981 | . . . . 5 β’ (π΄ β β+ β π΄ β β) | |
2 | 1 | adantr 481 | . . . 4 β’ ((π΄ β β+ β§ 1 < π΄) β π΄ β β) |
3 | 2 | resincld 16085 | . . 3 β’ ((π΄ β β+ β§ 1 < π΄) β (sinβπ΄) β β) |
4 | 1red 11214 | . . 3 β’ ((π΄ β β+ β§ 1 < π΄) β 1 β β) | |
5 | sinbnd 16122 | . . . . . 6 β’ (π΄ β β β (-1 β€ (sinβπ΄) β§ (sinβπ΄) β€ 1)) | |
6 | 5 | simprd 496 | . . . . 5 β’ (π΄ β β β (sinβπ΄) β€ 1) |
7 | 1, 6 | syl 17 | . . . 4 β’ (π΄ β β+ β (sinβπ΄) β€ 1) |
8 | 7 | adantr 481 | . . 3 β’ ((π΄ β β+ β§ 1 < π΄) β (sinβπ΄) β€ 1) |
9 | simpr 485 | . . 3 β’ ((π΄ β β+ β§ 1 < π΄) β 1 < π΄) | |
10 | 3, 4, 2, 8, 9 | lelttrd 11371 | . 2 β’ ((π΄ β β+ β§ 1 < π΄) β (sinβπ΄) < π΄) |
11 | df-3an 1089 | . . . 4 β’ ((π΄ β β β§ 0 < π΄ β§ π΄ β€ 1) β ((π΄ β β β§ 0 < π΄) β§ π΄ β€ 1)) | |
12 | 0xr 11260 | . . . . 5 β’ 0 β β* | |
13 | 1re 11213 | . . . . 5 β’ 1 β β | |
14 | elioc2 13386 | . . . . 5 β’ ((0 β β* β§ 1 β β) β (π΄ β (0(,]1) β (π΄ β β β§ 0 < π΄ β§ π΄ β€ 1))) | |
15 | 12, 13, 14 | mp2an 690 | . . . 4 β’ (π΄ β (0(,]1) β (π΄ β β β§ 0 < π΄ β§ π΄ β€ 1)) |
16 | elrp 12975 | . . . . 5 β’ (π΄ β β+ β (π΄ β β β§ 0 < π΄)) | |
17 | 16 | anbi1i 624 | . . . 4 β’ ((π΄ β β+ β§ π΄ β€ 1) β ((π΄ β β β§ 0 < π΄) β§ π΄ β€ 1)) |
18 | 11, 15, 17 | 3bitr4i 302 | . . 3 β’ (π΄ β (0(,]1) β (π΄ β β+ β§ π΄ β€ 1)) |
19 | sin01bnd 16127 | . . . 4 β’ (π΄ β (0(,]1) β ((π΄ β ((π΄β3) / 3)) < (sinβπ΄) β§ (sinβπ΄) < π΄)) | |
20 | 19 | simprd 496 | . . 3 β’ (π΄ β (0(,]1) β (sinβπ΄) < π΄) |
21 | 18, 20 | sylbir 234 | . 2 β’ ((π΄ β β+ β§ π΄ β€ 1) β (sinβπ΄) < π΄) |
22 | 1red 11214 | . 2 β’ (π΄ β β+ β 1 β β) | |
23 | 10, 21, 22, 1 | ltlecasei 11321 | 1 β’ (π΄ β β+ β (sinβπ΄) < π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 β wcel 2106 class class class wbr 5148 βcfv 6543 (class class class)co 7408 βcr 11108 0cc0 11109 1c1 11110 β*cxr 11246 < clt 11247 β€ cle 11248 β cmin 11443 -cneg 11444 / cdiv 11870 3c3 12267 β+crp 12973 (,]cioc 13324 βcexp 14026 sincsin 16006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12974 df-ioc 13328 df-ico 13329 df-fz 13484 df-fzo 13627 df-fl 13756 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15632 df-ef 16010 df-sin 16012 df-cos 16013 |
This theorem is referenced by: pigt3 26026 basellem8 26589 |
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