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Mirrors > Home > MPE Home > Th. List > ibl0 | Structured version Visualization version GIF version |
Description: The zero function is integrable on any measurable set. (Unlike iblconst 25671, this does not require š“ to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
ibl0 | ⢠(š“ ā dom vol ā (š“ Ć {0}) ā šæ1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11204 | . . 3 ⢠0 ā ā | |
2 | mbfconst 25486 | . . 3 ⢠((š“ ā dom vol ā§ 0 ā ā) ā (š“ Ć {0}) ā MblFn) | |
3 | 1, 2 | mpan2 688 | . 2 ⢠(š“ ā dom vol ā (š“ Ć {0}) ā MblFn) |
4 | ax-icn 11166 | . . . . . . . 8 ⢠i ā ā | |
5 | ine0 11647 | . . . . . . . 8 ⢠i ā 0 | |
6 | elfzelz 13499 | . . . . . . . . 9 ⢠(š ā (0...3) ā š ā ā¤) | |
7 | 6 | ad2antlr 724 | . . . . . . . 8 ⢠(((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā š ā ā¤) |
8 | expclz 14048 | . . . . . . . . 9 ⢠((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (iāš) ā ā) | |
9 | expne0i 14058 | . . . . . . . . 9 ⢠((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (iāš) ā 0) | |
10 | 8, 9 | div0d 11987 | . . . . . . . 8 ⢠((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (0 / (iāš)) = 0) |
11 | 4, 5, 7, 10 | mp3an12i 1461 | . . . . . . 7 ⢠(((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (0 / (iāš)) = 0) |
12 | 11 | fveq2d 6886 | . . . . . 6 ⢠(((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (āā(0 / (iāš))) = (āā0)) |
13 | re0 15097 | . . . . . 6 ⢠(āā0) = 0 | |
14 | 12, 13 | eqtrdi 2780 | . . . . 5 ⢠(((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (āā(0 / (iāš))) = 0) |
15 | 14 | itgvallem3 25639 | . . . 4 ⢠((š“ ā dom vol ā§ š ā (0...3)) ā (ā«2ā(š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) = 0) |
16 | 0re 11214 | . . . 4 ⢠0 ā ā | |
17 | 15, 16 | eqeltrdi 2833 | . . 3 ⢠((š“ ā dom vol ā§ š ā (0...3)) ā (ā«2ā(š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā) |
18 | 17 | ralrimiva 3138 | . 2 ⢠(š“ ā dom vol ā āš ā (0...3)(ā«2ā(š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā) |
19 | eqidd 2725 | . . 3 ⢠(š“ ā dom vol ā (š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0)) = (š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) | |
20 | eqidd 2725 | . . 3 ⢠((š“ ā dom vol ā§ š„ ā š“) ā (āā(0 / (iāš))) = (āā(0 / (iāš)))) | |
21 | c0ex 11206 | . . . . 5 ⢠0 ā V | |
22 | 21 | fconst 6768 | . . . 4 ⢠(š“ Ć {0}):š“ā¶{0} |
23 | fdm 6717 | . . . 4 ⢠((š“ Ć {0}):š“ā¶{0} ā dom (š“ Ć {0}) = š“) | |
24 | 22, 23 | mp1i 13 | . . 3 ⢠(š“ ā dom vol ā dom (š“ Ć {0}) = š“) |
25 | 21 | fvconst2 7198 | . . . 4 ⢠(š„ ā š“ ā ((š“ Ć {0})āš„) = 0) |
26 | 25 | adantl 481 | . . 3 ⢠((š“ ā dom vol ā§ š„ ā š“) ā ((š“ Ć {0})āš„) = 0) |
27 | 19, 20, 24, 26 | isibl 25619 | . 2 ⢠(š“ ā dom vol ā ((š“ Ć {0}) ā šæ1 ā ((š“ Ć {0}) ā MblFn ā§ āš ā (0...3)(ā«2ā(š„ ā ā ⦠if((š„ ā š“ ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā))) |
28 | 3, 18, 27 | mpbir2and 710 | 1 ⢠(š“ ā dom vol ā (š“ Ć {0}) ā šæ1) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā§ wa 395 ā§ w3a 1084 = wceq 1533 ā wcel 2098 ā wne 2932 āwral 3053 ifcif 4521 {csn 4621 class class class wbr 5139 ⦠cmpt 5222 Ć cxp 5665 dom cdm 5667 ā¶wf 6530 ācfv 6534 (class class class)co 7402 ācc 11105 ācr 11106 0cc0 11107 ici 11109 ⤠cle 11247 / cdiv 11869 3c3 12266 ā¤cz 12556 ...cfz 13482 ācexp 14025 ācre 15042 volcvol 25316 MblFncmbf 25467 ā«2citg2 25469 šæ1cibl 25470 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-disj 5105 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-n0 12471 df-z 12557 df-uz 12821 df-q 12931 df-rp 12973 df-xadd 13091 df-ioo 13326 df-ico 13328 df-icc 13329 df-fz 13483 df-fzo 13626 df-fl 13755 df-seq 13965 df-exp 14026 df-hash 14289 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-clim 15430 df-sum 15631 df-xmet 21223 df-met 21224 df-ovol 25317 df-vol 25318 df-mbf 25472 df-itg1 25473 df-itg2 25474 df-ibl 25475 df-0p 25523 |
This theorem is referenced by: itgge0 25664 itgfsum 25680 bddiblnc 25695 |
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