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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 13428 | . . 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 25468 | . . 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 13424 | . . . . . . 7 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β β) | |
| 6 | ax-resscn 11181 | . . . . . . 7 β’ β β β | |
| 7 | 5, 6 | sstrdi 3990 | . . . . . 6 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β β) |
| 8 | 7 | sselda 3978 | . . . . 5 β’ (((π΄ β β β§ π΅ β β) β§ π₯ β (π΄[,]π΅)) β π₯ β β) |
| 9 | 8 | 3adantl3 1166 | . . . 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 16092 | . . 3 β’ (((π΄ β β β§ π΅ β β β§ π β β0) β§ π₯ β (π΄[,]π΅)) β (sinβπ₯) β β) |
| 11 | simpl3 1191 | . . 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 14128 | . 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 45252 | . 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 25708 | 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 395 β§ w3a 1085 β wcel 2099 β wss 3944 β¦ cmpt 5225 dom cdm 5672 βcfv 6542 (class class class)co 7414 βcc 11122 βcr 11123 β0cn0 12488 (,)cioo 13342 [,]cicc 13345 βcexp 14044 sincsin 16025 volcvol 25366 πΏ1cibl 25520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cc 10444 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-ofr 7678 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8716 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15651 df-ef 16029 df-sin 16031 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-mulg 19008 df-cntz 19252 df-cmn 19721 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-cmp 23265 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-ovol 25367 df-vol 25368 df-mbf 25522 df-itg1 25523 df-itg2 25524 df-ibl 25525 df-0p 25573 df-limc 25769 df-dv 25770 |
| This theorem is referenced by: wallispilem1 45366 |
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