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Mirrors > Home > MPE Home > Th. List > cniccibl | Structured version Visualization version GIF version |
Description: A continuous function on a closed bounded interval is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.) |
Ref | Expression |
---|---|
cniccibl | β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β πΏ1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccmbl 25523 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β dom vol) | |
2 | cnmbf 25616 | . . 3 β’ (((π΄[,]π΅) β dom vol β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β MblFn) | |
3 | 1, 2 | stoic3 1770 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β MblFn) |
4 | simp3 1135 | . . . . 5 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β ((π΄[,]π΅)βcnββ)) | |
5 | cncff 24841 | . . . . 5 β’ (πΉ β ((π΄[,]π΅)βcnββ) β πΉ:(π΄[,]π΅)βΆβ) | |
6 | fdm 6736 | . . . . 5 β’ (πΉ:(π΄[,]π΅)βΆβ β dom πΉ = (π΄[,]π΅)) | |
7 | 4, 5, 6 | 3syl 18 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β dom πΉ = (π΄[,]π΅)) |
8 | 7 | fveq2d 6906 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) = (volβ(π΄[,]π΅))) |
9 | iccvolcl 25524 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (volβ(π΄[,]π΅)) β β) | |
10 | 9 | 3adant3 1129 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβ(π΄[,]π΅)) β β) |
11 | 8, 10 | eqeltrd 2829 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) β β) |
12 | cniccbdd 25418 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β βπ₯ β β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯) | |
13 | 7 | raleqdv 3323 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯ β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯)) |
14 | 13 | rexbidv 3176 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯ β βπ₯ β β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯)) |
15 | 12, 14 | mpbird 256 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) |
16 | bddibl 25797 | . 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) | |
17 | 3, 11, 15, 16 | syl3anc 1368 | 1 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β πΏ1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 βwrex 3067 class class class wbr 5152 dom cdm 5682 βΆwf 6549 βcfv 6553 (class class class)co 7426 βcc 11146 βcr 11147 β€ cle 11289 [,]cicc 13369 abscabs 15223 βcnβccncf 24824 volcvol 25420 MblFncmbf 25571 πΏ1cibl 25574 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cc 10468 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-addf 11227 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-disj 5118 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-ofr 7693 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-omul 8500 df-er 8733 df-map 8855 df-pm 8856 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-fi 9444 df-sup 9475 df-inf 9476 df-oi 9543 df-dju 9934 df-card 9972 df-acn 9975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13134 df-xadd 13135 df-xmul 13136 df-ioo 13370 df-ioc 13371 df-ico 13372 df-icc 13373 df-fz 13527 df-fzo 13670 df-fl 13799 df-mod 13877 df-seq 14009 df-exp 14069 df-hash 14332 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-limsup 15457 df-clim 15474 df-rlim 15475 df-sum 15675 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-hom 17266 df-cco 17267 df-rest 17413 df-topn 17414 df-0g 17432 df-gsum 17433 df-topgen 17434 df-pt 17435 df-prds 17438 df-xrs 17493 df-qtop 17498 df-imas 17499 df-xps 17501 df-mre 17575 df-mrc 17576 df-acs 17578 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-submnd 18750 df-mulg 19038 df-cntz 19282 df-cmn 19751 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-top 22824 df-topon 22841 df-topsp 22863 df-bases 22877 df-cn 23159 df-cnp 23160 df-cmp 23319 df-tx 23494 df-hmeo 23687 df-xms 24254 df-ms 24255 df-tms 24256 df-cncf 24826 df-ovol 25421 df-vol 25422 df-mbf 25576 df-itg1 25577 df-itg2 25578 df-ibl 25579 df-0p 25627 |
This theorem is referenced by: itgsubstlem 26011 iblidicc 34265 fdvposlt 34272 fdvposle 34274 circlemeth 34313 arearect 42692 areaquad 42693 lhe4.4ex1a 43815 itgsin0pilem1 45385 ibliccsinexp 45386 itgsinexplem1 45389 itgsinexp 45390 itgcoscmulx 45404 itgsincmulx 45409 itgioocnicc 45412 iblcncfioo 45413 dirkeritg 45537 fourierdlem95 45636 sqwvfoura 45663 sqwvfourb 45664 etransclem18 45687 |
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