MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismbf Structured version   Unicode version

Theorem ismbf 19514
Description: The predicate " F is a measurable function". A function is measurable iff the preimages of all open intervals are measurable sets in the sense of ismbl 19414. (Contributed by Mario Carneiro, 17-Jun-2014.)
Assertion
Ref Expression
ismbf  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem ismbf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mbfdm 19512 . . 3  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
2 fdm 5587 . . . 4  |-  ( F : A --> RR  ->  dom 
F  =  A )
32eleq1d 2501 . . 3  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
41, 3syl5ib 211 . 2  |-  ( F : A --> RR  ->  ( F  e. MblFn  ->  A  e. 
dom  vol ) )
5 ioomax 10977 . . . . 5  |-  (  -oo (,) 
+oo )  =  RR
6 ioorebas 10998 . . . . 5  |-  (  -oo (,) 
+oo )  e.  ran  (,)
75, 6eqeltrri 2506 . . . 4  |-  RR  e.  ran  (,)
8 imaeq2 5191 . . . . . 6  |-  ( x  =  RR  ->  ( `' F " x )  =  ( `' F " RR ) )
98eleq1d 2501 . . . . 5  |-  ( x  =  RR  ->  (
( `' F "
x )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
109rspcv 3040 . . . 4  |-  ( RR  e.  ran  (,)  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  ( `' F " RR )  e.  dom  vol ) )
117, 10ax-mp 8 . . 3  |-  ( A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol  ->  ( `' F " RR )  e.  dom  vol )
12 fimacnv 5854 . . . 4  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1312eleq1d 2501 . . 3  |-  ( F : A --> RR  ->  ( ( `' F " RR )  e.  dom  vol  <->  A  e.  dom  vol )
)
1411, 13syl5ib 211 . 2  |-  ( F : A --> RR  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  A  e.  dom  vol ) )
15 ffvelrn 5860 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
1615adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  RR )
1716rered 12021 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Re `  ( F `  x ) )  =  ( F `  x
) )
1817mpteq2dva 4287 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Re `  ( F `  x )
) )  =  ( x  e.  A  |->  ( F `  x ) ) )
1916recnd 9106 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 simpl 444 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F : A --> RR )
2120feqmptd 5771 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
22 ref 11909 . . . . . . . . . . . . . 14  |-  Re : CC
--> RR
2322a1i 11 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re : CC --> RR )
2423feqmptd 5771 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re  =  ( y  e.  CC  |->  ( Re
`  y ) ) )
25 fveq2 5720 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
Re `  y )  =  ( Re `  ( F `  x ) ) )
2619, 21, 24, 25fmptco 5893 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Re  o.  F
)  =  ( x  e.  A  |->  ( Re
`  ( F `  x ) ) ) )
2718, 26, 213eqtr4rd 2478 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( Re  o.  F ) )
2827cnveqd 5040 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' F  =  `' ( Re  o.  F
) )
2928imaeq1d 5194 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' F "
x )  =  ( `' ( Re  o.  F ) " x
) )
3029eleq1d 2501 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
31 imf 11910 . . . . . . . . . . . . . . . 16  |-  Im : CC
--> RR
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im : CC --> RR )
3332feqmptd 5771 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im  =  ( y  e.  CC  |->  ( Im
`  y ) ) )
34 fveq2 5720 . . . . . . . . . . . . . 14  |-  ( y  =  ( F `  x )  ->  (
Im `  y )  =  ( Im `  ( F `  x ) ) )
3519, 21, 33, 34fmptco 5893 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  ( Im
`  ( F `  x ) ) ) )
3616reim0d 12022 . . . . . . . . . . . . . 14  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Im `  ( F `  x ) )  =  0 )
3736mpteq2dva 4287 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Im `  ( F `  x )
) )  =  ( x  e.  A  |->  0 ) )
3835, 37eqtrd 2467 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  0 ) )
39 fconstmpt 4913 . . . . . . . . . . . 12  |-  ( A  X.  { 0 } )  =  ( x  e.  A  |->  0 )
4038, 39syl6eqr 2485 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( A  X.  { 0 } ) )
4140cnveqd 5040 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' ( Im  o.  F )  =  `' ( A  X.  { 0 } ) )
4241imaeq1d 5194 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  =  ( `' ( A  X.  { 0 } )
" x ) )
43 simpr 448 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  A  e.  dom  vol )
44 0re 9083 . . . . . . . . . 10  |-  0  e.  RR
45 mbfconstlem 19513 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  0  e.  RR )  ->  ( `' ( A  X.  { 0 } ) " x
)  e.  dom  vol )
4643, 44, 45sylancl 644 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( A  X.  { 0 } ) " x )  e.  dom  vol )
4742, 46eqeltrd 2509 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  e.  dom  vol )
4847biantrud 494 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  <->  ( ( `' ( Re  o.  F
) " x )  e.  dom  vol  /\  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) ) )
4930, 48bitrd 245 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( ( `' ( Re  o.  F ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  F )
" x )  e. 
dom  vol ) ) )
5049ralbidv 2717 . . . . 5  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( `' F " x )  e.  dom  vol  <->  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) )
51 ax-resscn 9039 . . . . . . . 8  |-  RR  C_  CC
52 fss 5591 . . . . . . . 8  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
5351, 52mpan2 653 . . . . . . 7  |-  ( F : A --> RR  ->  F : A --> CC )
54 mblss 19419 . . . . . . 7  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
55 cnex 9063 . . . . . . . 8  |-  CC  e.  _V
56 reex 9073 . . . . . . . 8  |-  RR  e.  _V
57 elpm2r 7026 . . . . . . . 8  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
5855, 56, 57mpanl12 664 . . . . . . 7  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
5953, 54, 58syl2an 464 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  e.  ( CC 
^pm  RR ) )
6059biantrurd 495 . . . . 5  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( ( `' ( Re  o.  F
) " x )  e.  dom  vol  /\  ( `' ( Im  o.  F ) " x
)  e.  dom  vol ) 
<->  ( F  e.  ( CC  ^pm  RR )  /\  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) ) )
6150, 60bitrd 245 . . . 4  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( A. x  e. 
ran  (,) ( `' F " x )  e.  dom  vol  <->  ( F  e.  ( CC 
^pm  RR )  /\  A. x  e.  ran  (,) (
( `' ( Re  o.  F ) "
x )  e.  dom  vol 
/\  ( `' ( Im  o.  F )
" x )  e. 
dom  vol ) ) ) )
62 ismbf1 19510 . . . 4  |-  ( F  e. MblFn 
<->  ( F  e.  ( CC  ^pm  RR )  /\  A. x  e.  ran  (,) ( ( `' ( Re  o.  F )
" x )  e. 
dom  vol  /\  ( `' ( Im  o.  F
) " x )  e.  dom  vol )
) )
6361, 62syl6rbbr 256 . . 3  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
6463ex 424 . 2  |-  ( F : A --> RR  ->  ( A  e.  dom  vol  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
) )
654, 14, 64pm5.21ndd 344 1  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   {csn 3806    e. cmpt 4258    X. cxp 4868   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    o. ccom 4874   -->wf 5442   ` cfv 5446  (class class class)co 6073    ^pm cpm 7011   CCcc 8980   RRcr 8981   0cc0 8982    +oocpnf 9109    -oocmnf 9110   (,)cioo 10908   Recre 11894   Imcim 11895   volcvol 19352  MblFncmbf 19498
This theorem is referenced by:  ismbfcn  19515  mbfima  19516  mbfid  19520  ismbfd  19524  mbfeqalem  19526  mbfres2  19529  mbfimaopnlem  19539  i1fd  19565  elmbfmvol2  24609  cnambfre  26245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-xadd 10703  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-xmet 16687  df-met 16688  df-ovol 19353  df-vol 19354  df-mbf 19504
  Copyright terms: Public domain W3C validator