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

Theorem ismbf 22665
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 22558. (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 22663 . . 3  |-  ( F  e. MblFn  ->  dom  F  e.  dom  vol )
2 fdm 5745 . . . 4  |-  ( F : A --> RR  ->  dom 
F  =  A )
32eleq1d 2533 . . 3  |-  ( F : A --> RR  ->  ( dom  F  e.  dom  vol  <->  A  e.  dom  vol )
)
41, 3syl5ib 227 . 2  |-  ( F : A --> RR  ->  ( F  e. MblFn  ->  A  e. 
dom  vol ) )
5 ioomax 11734 . . . . 5  |-  ( -oo (,) +oo )  =  RR
6 ioorebas 11761 . . . . 5  |-  ( -oo (,) +oo )  e.  ran  (,)
75, 6eqeltrri 2546 . . . 4  |-  RR  e.  ran  (,)
8 imaeq2 5170 . . . . . 6  |-  ( x  =  RR  ->  ( `' F " x )  =  ( `' F " RR ) )
98eleq1d 2533 . . . . 5  |-  ( x  =  RR  ->  (
( `' F "
x )  e.  dom  vol  <->  ( `' F " RR )  e.  dom  vol )
)
109rspcv 3132 . . . 4  |-  ( RR  e.  ran  (,)  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  ( `' F " RR )  e.  dom  vol ) )
117, 10ax-mp 5 . . 3  |-  ( A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol  ->  ( `' F " RR )  e.  dom  vol )
12 fimacnv 6027 . . . 4  |-  ( F : A --> RR  ->  ( `' F " RR )  =  A )
1312eleq1d 2533 . . 3  |-  ( F : A --> RR  ->  ( ( `' F " RR )  e.  dom  vol  <->  A  e.  dom  vol )
)
1411, 13syl5ib 227 . 2  |-  ( F : A --> RR  ->  ( A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol 
->  A  e.  dom  vol ) )
15 ffvelrn 6035 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
1615adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  RR )
1716rered 13364 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Re `  ( F `  x ) )  =  ( F `  x
) )
1817mpteq2dva 4482 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Re `  ( F `  x )
) )  =  ( x  e.  A  |->  ( F `  x ) ) )
1916recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 simpl 464 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F : A --> RR )
2120feqmptd 5932 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
22 ref 13252 . . . . . . . . . . . . . 14  |-  Re : CC
--> RR
2322a1i 11 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re : CC --> RR )
2423feqmptd 5932 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Re  =  ( y  e.  CC  |->  ( Re
`  y ) ) )
25 fveq2 5879 . . . . . . . . . . . 12  |-  ( y  =  ( F `  x )  ->  (
Re `  y )  =  ( Re `  ( F `  x ) ) )
2619, 21, 24, 25fmptco 6072 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Re  o.  F
)  =  ( x  e.  A  |->  ( Re
`  ( F `  x ) ) ) )
2718, 26, 213eqtr4rd 2516 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  =  ( Re  o.  F ) )
2827cnveqd 5015 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' F  =  `' ( Re  o.  F
) )
2928imaeq1d 5173 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' F "
x )  =  ( `' ( Re  o.  F ) " x
) )
3029eleq1d 2533 . . . . . . 7  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( ( `' F " x )  e.  dom  vol  <->  ( `' ( Re  o.  F ) " x
)  e.  dom  vol ) )
31 imf 13253 . . . . . . . . . . . . . . . 16  |-  Im : CC
--> RR
3231a1i 11 . . . . . . . . . . . . . . 15  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im : CC --> RR )
3332feqmptd 5932 . . . . . . . . . . . . . 14  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  Im  =  ( y  e.  CC  |->  ( Im
`  y ) ) )
34 fveq2 5879 . . . . . . . . . . . . . 14  |-  ( y  =  ( F `  x )  ->  (
Im `  y )  =  ( Im `  ( F `  x ) ) )
3519, 21, 33, 34fmptco 6072 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  ( Im
`  ( F `  x ) ) ) )
3616reim0d 13365 . . . . . . . . . . . . . 14  |-  ( ( ( F : A --> RR  /\  A  e.  dom  vol )  /\  x  e.  A )  ->  (
Im `  ( F `  x ) )  =  0 )
3736mpteq2dva 4482 . . . . . . . . . . . . 13  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( x  e.  A  |->  ( Im `  ( F `  x )
) )  =  ( x  e.  A  |->  0 ) )
3835, 37eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( x  e.  A  |->  0 ) )
39 fconstmpt 4883 . . . . . . . . . . . 12  |-  ( A  X.  { 0 } )  =  ( x  e.  A  |->  0 )
4038, 39syl6eqr 2523 . . . . . . . . . . 11  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( Im  o.  F
)  =  ( A  X.  { 0 } ) )
4140cnveqd 5015 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  `' ( Im  o.  F )  =  `' ( A  X.  { 0 } ) )
4241imaeq1d 5173 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  =  ( `' ( A  X.  { 0 } )
" x ) )
43 simpr 468 . . . . . . . . . 10  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  A  e.  dom  vol )
44 0re 9661 . . . . . . . . . 10  |-  0  e.  RR
45 mbfconstlem 22664 . . . . . . . . . 10  |-  ( ( A  e.  dom  vol  /\  0  e.  RR )  ->  ( `' ( A  X.  { 0 } ) " x
)  e.  dom  vol )
4643, 44, 45sylancl 675 . . . . . . . . 9  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( A  X.  { 0 } ) " x )  e.  dom  vol )
4742, 46eqeltrd 2549 . . . . . . . 8  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( `' ( Im  o.  F ) "
x )  e.  dom  vol )
4847biantrud 515 . . . . . . 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 261 . . . . . 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 2829 . . . . 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 9614 . . . . . . . 8  |-  RR  C_  CC
52 fss 5749 . . . . . . . 8  |-  ( ( F : A --> RR  /\  RR  C_  CC )  ->  F : A --> CC )
5351, 52mpan2 685 . . . . . . 7  |-  ( F : A --> RR  ->  F : A --> CC )
54 mblss 22563 . . . . . . 7  |-  ( A  e.  dom  vol  ->  A 
C_  RR )
55 cnex 9638 . . . . . . . 8  |-  CC  e.  _V
56 reex 9648 . . . . . . . 8  |-  RR  e.  _V
57 elpm2r 7507 . . . . . . . 8  |-  ( ( ( CC  e.  _V  /\  RR  e.  _V )  /\  ( F : A --> CC  /\  A  C_  RR ) )  ->  F  e.  ( CC  ^pm  RR ) )
5855, 56, 57mpanl12 696 . . . . . . 7  |-  ( ( F : A --> CC  /\  A  C_  RR )  ->  F  e.  ( CC  ^pm 
RR ) )
5953, 54, 58syl2an 485 . . . . . 6  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  F  e.  ( CC 
^pm  RR ) )
6059biantrurd 516 . . . . 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 261 . . . 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 22661 . . . 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 272 . . 3  |-  ( ( F : A --> RR  /\  A  e.  dom  vol )  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
)
6463ex 441 . 2  |-  ( F : A --> RR  ->  ( A  e.  dom  vol  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F " x )  e.  dom  vol )
) )
654, 14, 64pm5.21ndd 361 1  |-  ( F : A --> RR  ->  ( F  e. MblFn  <->  A. x  e.  ran  (,) ( `' F "
x )  e.  dom  vol ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   _Vcvv 3031    C_ wss 3390   {csn 3959    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^pm cpm 7491   CCcc 9555   RRcr 9556   0cc0 9557   +oocpnf 9690   -oocmnf 9691   (,)cioo 11660   Recre 13237   Imcim 13238   volcvol 22493  MblFncmbf 22651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-xmet 19040  df-met 19041  df-ovol 22494  df-vol 22496  df-mbf 22656
This theorem is referenced by:  ismbfcn  22666  mbfima  22667  mbfid  22671  ismbfd  22675  mbfeqalem  22677  mbfres2  22680  mbfimaopnlem  22690  i1fd  22718  elmbfmvol2  29162  cnambfre  32053  mbf0  37931
  Copyright terms: Public domain W3C validator