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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 25450 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β dom vol) | |
2 | cnmbf 25543 | . . 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 24768 | . . . . 5 β’ (πΉ β ((π΄[,]π΅)βcnββ) β πΉ:(π΄[,]π΅)βΆβ) | |
6 | fdm 6720 | . . . . 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 6889 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) = (volβ(π΄[,]π΅))) |
9 | iccvolcl 25451 | . . . 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 2827 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) β β) |
12 | cniccbdd 25345 | . . 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 3319 | . . . 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 3172 | . . 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 257 | . 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 25724 | . 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 3055 βwrex 3064 class class class wbr 5141 dom cdm 5669 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcc 11110 βcr 11111 β€ cle 11253 [,]cicc 13333 abscabs 15187 βcnβccncf 24751 volcvol 25347 MblFncmbf 25498 πΏ1cibl 25501 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-ovol 25348 df-vol 25349 df-mbf 25503 df-itg1 25504 df-itg2 25505 df-ibl 25506 df-0p 25554 |
This theorem is referenced by: itgsubstlem 25938 iblidicc 34133 fdvposlt 34140 fdvposle 34142 circlemeth 34181 arearect 42537 areaquad 42538 lhe4.4ex1a 43661 itgsin0pilem1 45235 ibliccsinexp 45236 itgsinexplem1 45239 itgsinexp 45240 itgcoscmulx 45254 itgsincmulx 45259 itgioocnicc 45262 iblcncfioo 45263 dirkeritg 45387 fourierdlem95 45486 sqwvfoura 45513 sqwvfourb 45514 etransclem18 45537 |
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