![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 25513 | . 2 ⢠ā ā MblFn | |
2 | fconstmpt 5731 | . . . . . . 7 ⢠(ā Ć {0}) = (š„ ā ā ⦠0) | |
3 | 2 | eqcomi 2735 | . . . . . 6 ⢠(š„ ā ā ⦠0) = (ā Ć {0}) |
4 | 3 | fveq2i 6887 | . . . . 5 ⢠(ā«2ā(š„ ā ā ⦠0)) = (ā«2ā(ā Ć {0})) |
5 | itg20 25617 | . . . . 5 ⢠(ā«2ā(ā Ć {0})) = 0 | |
6 | 4, 5 | eqtri 2754 | . . . 4 ⢠(ā«2ā(š„ ā ā ⦠0)) = 0 |
7 | 0re 11217 | . . . 4 ⢠0 ā ā | |
8 | 6, 7 | eqeltri 2823 | . . 3 ⢠(ā«2ā(š„ ā ā ⦠0)) ā ā |
9 | 8 | rgenw 3059 | . 2 ā¢ āš ā (0...3)(ā«2ā(š„ ā ā ⦠0)) ā ā |
10 | noel 4325 | . . . . . . . . 9 ⢠¬ š„ ā ā | |
11 | 10 | intnanr 487 | . . . . . . . 8 ⢠¬ (š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))) |
12 | 11 | iffalsei 4533 | . . . . . . 7 ⢠if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0) = 0 |
13 | 12 | eqcomi 2735 | . . . . . 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 5241 | . . . 4 ⢠(⤠ā (š„ ā ā ⦠0) = (š„ ā ā ⦠if((š„ ā ā ā§ 0 ⤠(āā(0 / (iāš)))), (āā(0 / (iāš))), 0))) |
16 | eqidd 2727 | . . . 4 ⢠((⤠⧠š„ ā ā ) ā (āā(0 / (iāš))) = (āā(0 / (iāš)))) | |
17 | dm0 5913 | . . . . 5 ⢠dom ā = ā | |
18 | 17 | a1i 11 | . . . 4 ⢠(⤠ā dom ā = ā ) |
19 | 10 | intnan 486 | . . . . 5 ⢠¬ (⤠⧠š„ ā ā ) |
20 | 19 | pm2.21i 119 | . . . 4 ⢠((⤠⧠š„ ā ā ) ā (ā āš„) = 0) |
21 | 15, 16, 18, 20 | isibl 25645 | . . 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 708 | 1 ⢠ā ā šæ1 |
Colors of variables: wff setvar class |
Syntax hints: ā wb 205 ā§ wa 395 = wceq 1533 ā¤wtru 1534 ā wcel 2098 āwral 3055 ā c0 4317 ifcif 4523 {csn 4623 class class class wbr 5141 ⦠cmpt 5224 Ć cxp 5667 dom cdm 5669 ācfv 6536 (class class class)co 7404 ācr 11108 0cc0 11109 ici 11111 ⤠cle 11250 / cdiv 11872 3c3 12269 ...cfz 13487 ācexp 14029 ācre 15047 MblFncmbf 25493 ā«2citg2 25495 šæ1cibl 25496 |
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 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
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-op 4630 df-uni 4903 df-int 4944 df-iun 4992 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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-sum 15636 df-xmet 21228 df-met 21229 df-ovol 25343 df-vol 25344 df-mbf 25498 df-itg1 25499 df-itg2 25500 df-ibl 25501 df-0p 25549 |
This theorem is referenced by: itgvol0 45238 |
Copyright terms: Public domain | W3C validator |