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Mirrors > Home > MPE Home > Th. List > ismbfcn2 | Structured version Visualization version GIF version |
Description: A complex function is measurable iff the real and imaginary components of the function are measurable. (Contributed by Mario Carneiro, 13-Aug-2014.) |
Ref | Expression |
---|---|
ismbfcn2.1 | β’ ((π β§ π₯ β π΄) β π΅ β β) |
Ref | Expression |
---|---|
ismbfcn2 | β’ (π β ((π₯ β π΄ β¦ π΅) β MblFn β ((π₯ β π΄ β¦ (ββπ΅)) β MblFn β§ (π₯ β π΄ β¦ (ββπ΅)) β MblFn))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbfcn2.1 | . . . 4 β’ ((π β§ π₯ β π΄) β π΅ β β) | |
2 | 1 | fmpttd 7118 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
3 | ismbfcn 25574 | . . 3 β’ ((π₯ β π΄ β¦ π΅):π΄βΆβ β ((π₯ β π΄ β¦ π΅) β MblFn β ((β β (π₯ β π΄ β¦ π΅)) β MblFn β§ (β β (π₯ β π΄ β¦ π΅)) β MblFn))) | |
4 | 2, 3 | syl 17 | . 2 β’ (π β ((π₯ β π΄ β¦ π΅) β MblFn β ((β β (π₯ β π΄ β¦ π΅)) β MblFn β§ (β β (π₯ β π΄ β¦ π΅)) β MblFn))) |
5 | ref 15089 | . . . . . 6 β’ β:ββΆβ | |
6 | 5 | a1i 11 | . . . . 5 β’ (π β β:ββΆβ) |
7 | 6, 1 | cofmpt 7135 | . . . 4 β’ (π β (β β (π₯ β π΄ β¦ π΅)) = (π₯ β π΄ β¦ (ββπ΅))) |
8 | 7 | eleq1d 2810 | . . 3 β’ (π β ((β β (π₯ β π΄ β¦ π΅)) β MblFn β (π₯ β π΄ β¦ (ββπ΅)) β MblFn)) |
9 | imf 15090 | . . . . . 6 β’ β:ββΆβ | |
10 | 9 | a1i 11 | . . . . 5 β’ (π β β:ββΆβ) |
11 | 10, 1 | cofmpt 7135 | . . . 4 β’ (π β (β β (π₯ β π΄ β¦ π΅)) = (π₯ β π΄ β¦ (ββπ΅))) |
12 | 11 | eleq1d 2810 | . . 3 β’ (π β ((β β (π₯ β π΄ β¦ π΅)) β MblFn β (π₯ β π΄ β¦ (ββπ΅)) β MblFn)) |
13 | 8, 12 | anbi12d 630 | . 2 β’ (π β (((β β (π₯ β π΄ β¦ π΅)) β MblFn β§ (β β (π₯ β π΄ β¦ π΅)) β MblFn) β ((π₯ β π΄ β¦ (ββπ΅)) β MblFn β§ (π₯ β π΄ β¦ (ββπ΅)) β MblFn))) |
14 | 4, 13 | bitrd 278 | 1 β’ (π β ((π₯ β π΄ β¦ π΅) β MblFn β ((π₯ β π΄ β¦ (ββπ΅)) β MblFn β§ (π₯ β π΄ β¦ (ββπ΅)) β MblFn))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 β¦ cmpt 5224 β ccom 5674 βΆwf 6537 βcfv 6541 βcc 11134 βcr 11135 βcre 15074 βcim 15075 MblFncmbf 25559 |
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 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-er 8721 df-map 8843 df-pm 8844 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-dju 9922 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-q 12961 df-rp 13005 df-xadd 13123 df-ioo 13358 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-xmet 21274 df-met 21275 df-ovol 25409 df-vol 25410 df-mbf 25564 |
This theorem is referenced by: mbfeqa 25588 mbfss 25591 mbfmulc2re 25593 mbfadd 25606 mbfmulc2 25608 mbflim 25613 mbfmul 25672 iblcn 25744 ibladd 25766 ibladdnc 37179 ftc1anclem2 37196 ftc1anclem5 37199 ftc1anclem6 37200 ftc1anclem8 37202 |
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