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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 25205, this does not require š“ to have finite measure.) (Contributed by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
ibl0 | ā¢ (š“ ā dom vol ā (š“ Ć {0}) ā šæ1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 11155 | . . 3 ā¢ 0 ā ā | |
2 | mbfconst 25020 | . . 3 ā¢ ((š“ ā dom vol ā§ 0 ā ā) ā (š“ Ć {0}) ā MblFn) | |
3 | 1, 2 | mpan2 690 | . 2 ā¢ (š“ ā dom vol ā (š“ Ć {0}) ā MblFn) |
4 | ax-icn 11118 | . . . . . . . 8 ā¢ i ā ā | |
5 | ine0 11598 | . . . . . . . 8 ā¢ i ā 0 | |
6 | elfzelz 13450 | . . . . . . . . 9 ā¢ (š ā (0...3) ā š ā ā¤) | |
7 | 6 | ad2antlr 726 | . . . . . . . 8 ā¢ (((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā š ā ā¤) |
8 | expclz 13999 | . . . . . . . . 9 ā¢ ((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (iāš) ā ā) | |
9 | expne0i 14009 | . . . . . . . . 9 ā¢ ((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (iāš) ā 0) | |
10 | 8, 9 | div0d 11938 | . . . . . . . 8 ā¢ ((i ā ā ā§ i ā 0 ā§ š ā ā¤) ā (0 / (iāš)) = 0) |
11 | 4, 5, 7, 10 | mp3an12i 1466 | . . . . . . 7 ā¢ (((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (0 / (iāš)) = 0) |
12 | 11 | fveq2d 6850 | . . . . . 6 ā¢ (((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (āā(0 / (iāš))) = (āā0)) |
13 | re0 15046 | . . . . . 6 ā¢ (āā0) = 0 | |
14 | 12, 13 | eqtrdi 2789 | . . . . 5 ā¢ (((š“ ā dom vol ā§ š ā (0...3)) ā§ š„ ā š“) ā (āā(0 / (iāš))) = 0) |
15 | 14 | itgvallem3 25173 | . . . 4 ā¢ ((š“ ā dom vol ā§ š ā (0...3)) ā (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) = 0) |
16 | 0re 11165 | . . . 4 ā¢ 0 ā ā | |
17 | 15, 16 | eqeltrdi 2842 | . . 3 ā¢ ((š“ ā dom vol ā§ š ā (0...3)) ā (ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā) |
18 | 17 | ralrimiva 3140 | . 2 ā¢ (š“ ā dom vol ā āš ā (0...3)(ā«2ā(š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā) |
19 | eqidd 2734 | . . 3 ā¢ (š“ ā dom vol ā (š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0)) = (š„ ā ā ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) | |
20 | eqidd 2734 | . . 3 ā¢ ((š“ ā dom vol ā§ š„ ā š“) ā (āā(0 / (iāš))) = (āā(0 / (iāš)))) | |
21 | c0ex 11157 | . . . . 5 ā¢ 0 ā V | |
22 | 21 | fconst 6732 | . . . 4 ā¢ (š“ Ć {0}):š“ā¶{0} |
23 | fdm 6681 | . . . 4 ā¢ ((š“ Ć {0}):š“ā¶{0} ā dom (š“ Ć {0}) = š“) | |
24 | 22, 23 | mp1i 13 | . . 3 ā¢ (š“ ā dom vol ā dom (š“ Ć {0}) = š“) |
25 | 21 | fvconst2 7157 | . . . 4 ā¢ (š„ ā š“ ā ((š“ Ć {0})āš„) = 0) |
26 | 25 | adantl 483 | . . 3 ā¢ ((š“ ā dom vol ā§ š„ ā š“) ā ((š“ Ć {0})āš„) = 0) |
27 | 19, 20, 24, 26 | isibl 25153 | . 2 ā¢ (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 ā¦ if((š„ ā š“ ā§ 0 ā¤ (āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) ā ā))) |
28 | 3, 18, 27 | mpbir2and 712 | 1 ā¢ (š“ ā dom vol ā (š“ Ć {0}) ā šæ1) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā§ wa 397 ā§ w3a 1088 = wceq 1542 ā wcel 2107 ā wne 2940 āwral 3061 ifcif 4490 {csn 4590 class class class wbr 5109 ā¦ cmpt 5192 Ć cxp 5635 dom cdm 5637 ā¶wf 6496 ācfv 6500 (class class class)co 7361 ācc 11057 ācr 11058 0cc0 11059 ici 11061 ā¤ cle 11198 / cdiv 11820 3c3 12217 ā¤cz 12507 ...cfz 13433 ācexp 13976 ācre 14991 volcvol 24850 MblFncmbf 25001 ā«2citg2 25003 šæ1cibl 25004 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-disj 5075 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-ofr 7622 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xadd 13042 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-sum 15580 df-xmet 20812 df-met 20813 df-ovol 24851 df-vol 24852 df-mbf 25006 df-itg1 25007 df-itg2 25008 df-ibl 25009 df-0p 25057 |
This theorem is referenced by: itgge0 25198 itgfsum 25214 bddiblnc 25229 |
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