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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblempty | Structured version Visualization version GIF version |
Description: The empty function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iblempty | ⢠ā ā šæ1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbf0 25583 | . 2 ⢠ā ā MblFn | |
2 | fconstmpt 5744 | . . . . . . 7 ⢠(ā Ć {0}) = (š„ ā ā ⦠0) | |
3 | 2 | eqcomi 2737 | . . . . . 6 ⢠(š„ ā ā ⦠0) = (ā Ć {0}) |
4 | 3 | fveq2i 6905 | . . . . 5 ⢠(ā«2ā(š„ ā ā ⦠0)) = (ā«2ā(ā Ć {0})) |
5 | itg20 25687 | . . . . 5 ⢠(ā«2ā(ā Ć {0})) = 0 | |
6 | 4, 5 | eqtri 2756 | . . . 4 ⢠(ā«2ā(š„ ā ā ⦠0)) = 0 |
7 | 0re 11254 | . . . 4 ⢠0 ā ā | |
8 | 6, 7 | eqeltri 2825 | . . 3 ⢠(ā«2ā(š„ ā ā ⦠0)) ā ā |
9 | 8 | rgenw 3062 | . 2 ā¢ āš ā (0...3)(ā«2ā(š„ ā ā ⦠0)) ā ā |
10 | noel 4334 | . . . . . . . . 9 ⢠¬ š„ ā ā | |
11 | 10 | intnanr 486 | . . . . . . . 8 ⢠¬ (š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))) |
12 | 11 | iffalsei 4542 | . . . . . . 7 ⢠if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0) = 0 |
13 | 12 | eqcomi 2737 | . . . . . 6 ⢠0 = if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0) |
14 | 13 | a1i 11 | . . . . 5 ⢠((⤠⧠š„ ā ā) ā 0 = if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0)) |
15 | 14 | mpteq2dva 5252 | . . . 4 ⢠(⤠ā (š„ ā ā ⦠0) = (š„ ā ā ⦠if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) |
16 | eqidd 2729 | . . . 4 ⢠((⤠⧠š„ ā ā ) ā (āā(0 / (iāš))) = (āā(0 / (iāš)))) | |
17 | dm0 5927 | . . . . 5 ⢠dom ā = ā | |
18 | 17 | a1i 11 | . . . 4 ⢠(⤠ā dom ā = ā ) |
19 | 10 | intnan 485 | . . . . 5 ⢠¬ (⤠⧠š„ ā ā ) |
20 | 19 | pm2.21i 119 | . . . 4 ⢠((⤠⧠š„ ā ā ) ā (ā āš„) = 0) |
21 | 15, 16, 18, 20 | isibl 25715 | . . 3 ⢠(⤠ā (ā ā šæ1 ā (ā ā MblFn ā§ āš ā (0...3)(ā«2ā(š„ ā ā ⦠0)) ā ā))) |
22 | 21 | mptru 1540 | . 2 ⢠(ā ā šæ1 ā (ā ā MblFn ā§ āš ā (0...3)(ā«2ā(š„ ā ā ⦠0)) ā ā)) |
23 | 1, 9, 22 | mpbir2an 709 | 1 ⢠ā ā šæ1 |
Colors of variables: wff setvar class |
Syntax hints: ā wb 205 ā§ wa 394 = wceq 1533 ā¤wtru 1534 ā wcel 2098 āwral 3058 ā c0 4326 ifcif 4532 {csn 4632 class class class wbr 5152 ⦠cmpt 5235 Ć cxp 5680 dom cdm 5682 ācfv 6553 (class class class)co 7426 ācr 11145 0cc0 11146 ici 11148 ⤠cle 11287 / cdiv 11909 3c3 12306 ...cfz 13524 ācexp 14066 ācre 15084 MblFncmbf 25563 ā«2citg2 25565 šæ1cibl 25566 |
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 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 |
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-op 4639 df-uni 4913 df-int 4954 df-iun 5002 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 7691 df-ofr 7692 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 df-itg2 25570 df-ibl 25571 df-0p 25619 |
This theorem is referenced by: itgvol0 45385 |
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