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Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iblioosinexp | Structured version Visualization version GIF version | ||
| Description: sin^n on an open integral is integrable. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
| Ref | Expression |
|---|---|
| iblioosinexp | β’ ((π΄ β β β§ π΅ β β β§ π β β0) β (π₯ β (π΄(,)π΅) β¦ ((sinβπ₯)βπ)) β πΏ1) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ioossicc 13437 | . . 3 β’ (π΄(,)π΅) β (π΄[,]π΅) | |
| 2 | 1 | a1i 11 | . 2 β’ ((π΄ β β β§ π΅ β β β§ π β β0) β (π΄(,)π΅) β (π΄[,]π΅)) |
| 3 | ioombl 25507 | . . 3 β’ (π΄(,)π΅) β dom vol | |
| 4 | 3 | a1i 11 | . 2 β’ ((π΄ β β β§ π΅ β β β§ π β β0) β (π΄(,)π΅) β dom vol) |
| 5 | iccssre 13433 | . . . . . . 7 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β β) | |
| 6 | ax-resscn 11190 | . . . . . . 7 β’ β β β | |
| 7 | 5, 6 | sstrdi 3986 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β β) |
| 8 | 7 | sselda 3973 | . . . . 5 β’ (((π΄ β β β§ π΅ β β) β§ π₯ β (π΄[,]π΅)) β π₯ β β) |
| 9 | 8 | 3adantl3 1165 | . . . 4 β’ (((π΄ β β β§ π΅ β β β§ π β β0) β§ π₯ β (π΄[,]π΅)) β π₯ β β) |
| 10 | 9 | sincld 16101 | . . 3 β’ (((π΄ β β β§ π΅ β β β§ π β β0) β§ π₯ β (π΄[,]π΅)) β (sinβπ₯) β β) |
| 11 | simpl3 1190 | . . 3 β’ (((π΄ β β β§ π΅ β β β§ π β β0) β§ π₯ β (π΄[,]π΅)) β π β β0) | |
| 12 | 10, 11 | expcld 14137 | . 2 β’ (((π΄ β β β§ π΅ β β β§ π β β0) β§ π₯ β (π΄[,]π΅)) β ((sinβπ₯)βπ) β β) |
| 13 | ibliccsinexp 45398 | . 2 β’ ((π΄ β β β§ π΅ β β β§ π β β0) β (π₯ β (π΄[,]π΅) β¦ ((sinβπ₯)βπ)) β πΏ1) | |
| 14 | 2, 4, 12, 13 | iblss 25747 | 1 β’ ((π΄ β β β§ π΅ β β β§ π β β0) β (π₯ β (π΄(,)π΅) β¦ ((sinβπ₯)βπ)) β πΏ1) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 β wcel 2098 β wss 3941 β¦ cmpt 5227 dom cdm 5673 βcfv 6543 (class class class)co 7413 βcc 11131 βcr 11132 β0cn0 12497 (,)cioo 13351 [,]cicc 13354 βcexp 14053 sincsin 16034 volcvol 25405 πΏ1cibl 25559 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cc 10453 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-sum 15660 df-ef 16038 df-sin 16040 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19023 df-cntz 19267 df-cmn 19736 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-ovol 25406 df-vol 25407 df-mbf 25561 df-itg1 25562 df-itg2 25563 df-ibl 25564 df-0p 25612 df-limc 25808 df-dv 25809 |
| This theorem is referenced by: wallispilem1 45512 |
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