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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnioobibld | Structured version Visualization version GIF version |
Description: A bounded, continuous function on an open bounded interval is integrable. The function must be bounded. For a counterexample, consider πΉ = (π₯ β (0(,)1) β¦ (1 / π₯)). See cniccibl 25358 for closed bounded intervals. (Contributed by Jon Pennant, 31-May-2019.) |
Ref | Expression |
---|---|
cnioobibld.1 | β’ (π β π΄ β β) |
cnioobibld.2 | β’ (π β π΅ β β) |
cnioobibld.3 | β’ (π β πΉ β ((π΄(,)π΅)βcnββ)) |
cnioobibld.4 | β’ (π β βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) |
Ref | Expression |
---|---|
cnioobibld | β’ (π β πΉ β πΏ1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioombl 25082 | . . 3 β’ (π΄(,)π΅) β dom vol | |
2 | cnioobibld.3 | . . 3 β’ (π β πΉ β ((π΄(,)π΅)βcnββ)) | |
3 | cnmbf 25176 | . . 3 β’ (((π΄(,)π΅) β dom vol β§ πΉ β ((π΄(,)π΅)βcnββ)) β πΉ β MblFn) | |
4 | 1, 2, 3 | sylancr 588 | . 2 β’ (π β πΉ β MblFn) |
5 | cncff 24409 | . . . . 5 β’ (πΉ β ((π΄(,)π΅)βcnββ) β πΉ:(π΄(,)π΅)βΆβ) | |
6 | fdm 6727 | . . . . 5 β’ (πΉ:(π΄(,)π΅)βΆβ β dom πΉ = (π΄(,)π΅)) | |
7 | 2, 5, 6 | 3syl 18 | . . . 4 β’ (π β dom πΉ = (π΄(,)π΅)) |
8 | 7 | fveq2d 6896 | . . 3 β’ (π β (volβdom πΉ) = (volβ(π΄(,)π΅))) |
9 | cnioobibld.1 | . . . 4 β’ (π β π΄ β β) | |
10 | cnioobibld.2 | . . . 4 β’ (π β π΅ β β) | |
11 | ioovolcl 25087 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (volβ(π΄(,)π΅)) β β) | |
12 | 9, 10, 11 | syl2anc 585 | . . 3 β’ (π β (volβ(π΄(,)π΅)) β β) |
13 | 8, 12 | eqeltrd 2834 | . 2 β’ (π β (volβdom πΉ) β β) |
14 | cnioobibld.4 | . 2 β’ (π β βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) | |
15 | bddibl 25357 | . 2 β’ ((πΉ β MblFn β§ (volβdom πΉ) β β β§ βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) β πΉ β πΏ1) | |
16 | 4, 13, 14, 15 | syl3anc 1372 | 1 β’ (π β πΉ β πΏ1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3062 βwrex 3071 class class class wbr 5149 dom cdm 5677 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 βcr 11109 β€ cle 11249 (,)cioo 13324 abscabs 15181 βcnβccncf 24392 volcvol 24980 MblFncmbf 25131 πΏ1cibl 25134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cn 22731 df-cnp 22732 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-ovol 24981 df-vol 24982 df-mbf 25136 df-itg1 25137 df-itg2 25138 df-ibl 25139 df-0p 25187 |
This theorem is referenced by: (None) |
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