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Theorem mbfi1fseq 22758
Description: A characterization of measurability in terms of simple functions (this is an if and only if for nonnegative functions, although we don't prove it). Any nonnegative measurable function is the limit of an increasing sequence of nonnegative simple functions. This proof is an example of a poor de Bruijn factor - the formalized proof is much longer than an average hand proof, which usually just describes the function  G and "leaves the details as an exercise to the reader". (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1  |-  ( ph  ->  F  e. MblFn )
mbfi1fseq.2  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
Assertion
Ref Expression
mbfi1fseq  |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
A. n  e.  NN  ( 0p  oR  <_  ( g `  n )  /\  (
g `  n )  oR  <_  ( g `
 ( n  + 
1 ) ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `
 n ) `  x ) )  ~~>  ( F `
 x ) ) )
Distinct variable groups:    g, n, x, F    ph, n, x
Allowed substitution hint:    ph( g)

Proof of Theorem mbfi1fseq
Dummy variables  j 
k  m  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfi1fseq.1 . 2  |-  ( ph  ->  F  e. MblFn )
2 mbfi1fseq.2 . 2  |-  ( ph  ->  F : RR --> ( 0 [,) +oo ) )
3 oveq2 6316 . . . . . 6  |-  ( j  =  k  ->  (
2 ^ j )  =  ( 2 ^ k ) )
43oveq2d 6324 . . . . 5  |-  ( j  =  k  ->  (
( F `  z
)  x.  ( 2 ^ j ) )  =  ( ( F `
 z )  x.  ( 2 ^ k
) ) )
54fveq2d 5883 . . . 4  |-  ( j  =  k  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  =  ( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) ) )
65, 3oveq12d 6326 . . 3  |-  ( j  =  k  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) )  =  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
7 fveq2 5879 . . . . . 6  |-  ( z  =  y  ->  ( F `  z )  =  ( F `  y ) )
87oveq1d 6323 . . . . 5  |-  ( z  =  y  ->  (
( F `  z
)  x.  ( 2 ^ k ) )  =  ( ( F `
 y )  x.  ( 2 ^ k
) ) )
98fveq2d 5883 . . . 4  |-  ( z  =  y  ->  ( |_ `  ( ( F `
 z )  x.  ( 2 ^ k
) ) )  =  ( |_ `  (
( F `  y
)  x.  ( 2 ^ k ) ) ) )
109oveq1d 6323 . . 3  |-  ( z  =  y  ->  (
( |_ `  (
( F `  z
)  x.  ( 2 ^ k ) ) )  /  ( 2 ^ k ) )  =  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
116, 10cbvmpt2v 6390 . 2  |-  ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) )  =  ( k  e.  NN ,  y  e.  RR  |->  ( ( |_
`  ( ( F `
 y )  x.  ( 2 ^ k
) ) )  / 
( 2 ^ k
) ) )
12 eleq1 2537 . . . . . 6  |-  ( y  =  x  ->  (
y  e.  ( -u m [,] m )  <->  x  e.  ( -u m [,] m
) ) )
13 oveq2 6316 . . . . . . . 8  |-  ( y  =  x  ->  (
m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y )  =  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) )
1413breq1d 4405 . . . . . . 7  |-  ( y  =  x  ->  (
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m  <->  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  m )
)
1514, 13ifbieq1d 3895 . . . . . 6  |-  ( y  =  x  ->  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y ) ,  m )  =  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  m ) )
1612, 15ifbieq1d 3895 . . . . 5  |-  ( y  =  x  ->  if ( y  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y ) ,  m ) ,  0 )  =  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  m ) ,  0 ) )
1716cbvmptv 4488 . . . 4  |-  ( y  e.  RR  |->  if ( y  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  m ) ,  0 ) )
18 negeq 9887 . . . . . . . 8  |-  ( m  =  k  ->  -u m  =  -u k )
19 id 22 . . . . . . . 8  |-  ( m  =  k  ->  m  =  k )
2018, 19oveq12d 6326 . . . . . . 7  |-  ( m  =  k  ->  ( -u m [,] m )  =  ( -u k [,] k ) )
2120eleq2d 2534 . . . . . 6  |-  ( m  =  k  ->  (
x  e.  ( -u m [,] m )  <->  x  e.  ( -u k [,] k
) ) )
22 oveq1 6315 . . . . . . . 8  |-  ( m  =  k  ->  (
m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  =  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) )
2322, 19breq12d 4408 . . . . . . 7  |-  ( m  =  k  ->  (
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m  <->  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  k )
)
2423, 22, 19ifbieq12d 3899 . . . . . 6  |-  ( m  =  k  ->  if ( ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m
( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  m )  =  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  k ) )
2521, 24ifbieq1d 3895 . . . . 5  |-  ( m  =  k  ->  if ( x  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x )  <_  m , 
( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  m
) ,  0 )  =  if ( x  e.  ( -u k [,] k ) ,  if ( ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_ 
k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x ) ,  k ) ,  0 ) )
2625mpteq2dv 4483 . . . 4  |-  ( m  =  k  ->  (
x  e.  RR  |->  if ( x  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  (
-u k [,] k
) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  k ) ,  0 ) ) )
2717, 26syl5eq 2517 . . 3  |-  ( m  =  k  ->  (
y  e.  RR  |->  if ( y  e.  (
-u m [,] m
) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) y ) ,  m ) ,  0 ) )  =  ( x  e.  RR  |->  if ( x  e.  (
-u k [,] k
) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN , 
z  e.  RR  |->  ( ( |_ `  (
( F `  z
)  x.  ( 2 ^ j ) ) )  /  ( 2 ^ j ) ) ) x ) ,  k ) ,  0 ) ) )
2827cbvmptv 4488 . 2  |-  ( m  e.  NN  |->  ( y  e.  RR  |->  if ( y  e.  ( -u m [,] m ) ,  if ( ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) y )  <_  m ,  ( m ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) y ) ,  m ) ,  0 ) ) )  =  ( k  e.  NN  |->  ( x  e.  RR  |->  if ( x  e.  ( -u k [,] k ) ,  if ( ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_ `  ( ( F `  z )  x.  (
2 ^ j ) ) )  /  (
2 ^ j ) ) ) x )  <_  k ,  ( k ( j  e.  NN ,  z  e.  RR  |->  ( ( |_
`  ( ( F `
 z )  x.  ( 2 ^ j
) ) )  / 
( 2 ^ j
) ) ) x ) ,  k ) ,  0 ) ) )
291, 2, 11, 28mbfi1fseqlem6 22757 1  |-  ( ph  ->  E. g ( g : NN --> dom  S.1  /\ 
A. n  e.  NN  ( 0p  oR  <_  ( g `  n )  /\  (
g `  n )  oR  <_  ( g `
 ( n  + 
1 ) ) )  /\  A. x  e.  RR  ( n  e.  NN  |->  ( ( g `
 n ) `  x ) )  ~~>  ( F `
 x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    /\ w3a 1007   E.wex 1671    e. wcel 1904   A.wral 2756   ifcif 3872   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    oRcofr 6549   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   +oocpnf 9690    <_ cle 9694   -ucneg 9881    / cdiv 10291   NNcn 10631   2c2 10681   [,)cico 11662   [,]cicc 11663   |_cfl 12059   ^cexp 12310    ~~> cli 13625  MblFncmbf 22651   S.1citg1 22652   0pc0p 22706
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-ofr 6551  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-fi 7943  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-xneg 11432  df-xadd 11433  df-xmul 11434  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-rlim 13630  df-sum 13830  df-rest 15399  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cmp 20479  df-ovol 22494  df-vol 22496  df-mbf 22656  df-itg1 22657  df-0p 22707
This theorem is referenced by:  mbfi1flimlem  22759  itg2add  22796
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