Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omsfval Structured version   Unicode version

Theorem omsfval 28981
Description: Value of the outer measure evaluated for a given set 
A. (Contributed by Thierry Arnoux, 15-Sep-2019.)
Assertion
Ref Expression
omsfval  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  ( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
Distinct variable groups:    x, y,
z, R    x, A, y, z    x, Q, y, z    x, V, y, z

Proof of Theorem omsfval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 simp2 1006 . . . 4  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  R : Q --> ( 0 [,] +oo ) )
2 simp1 1005 . . . 4  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  Q  e.  V
)
3 fex 6144 . . . 4  |-  ( ( R : Q --> ( 0 [,] +oo )  /\  Q  e.  V )  ->  R  e.  _V )
41, 2, 3syl2anc 665 . . 3  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  R  e.  _V )
5 omsval 28980 . . 3  |-  ( R  e.  _V  ->  (toOMeas `  R )  =  ( a  e.  ~P U. dom  R  |->  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
64, 5syl 17 . 2  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  (toOMeas `  R )  =  ( a  e. 
~P U. dom  R  |->  sup ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
7 simpr 462 . . . . . . . 8  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  a  =  A )
87sseq1d 3488 . . . . . . 7  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  ( a  C_ 
U. z  <->  A  C_  U. z
) )
98anbi1d 709 . . . . . 6  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  ( (
a  C_  U. z  /\  z  ~<_  om )  <->  ( A  C_  U. z  /\  z  ~<_  om )
) )
109rabbidv 3070 . . . . 5  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  { z  e.  ~P dom  R  | 
( a  C_  U. z  /\  z  ~<_  om ) }  =  { z  e.  ~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) } )
1110mpteq1d 4498 . . . 4  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  ( x  e.  { z  e.  ~P dom  R  |  ( a 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) )  =  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) )
1211rneqd 5073 . . 3  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  ran  ( x  e.  { z  e. 
~P dom  R  | 
( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) )  =  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) )
1312supeq1d 7957 . 2  |-  ( ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  /\  a  =  A )  ->  sup ( ran  ( x  e.  {
z  e.  ~P dom  R  |  ( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  )  =  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
14 simp3 1007 . . . 4  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  A  C_  U. Q
)
15 fdm 5741 . . . . . 6  |-  ( R : Q --> ( 0 [,] +oo )  ->  dom  R  =  Q )
16153ad2ant2 1027 . . . . 5  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  dom  R  =  Q )
1716unieqd 4223 . . . 4  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  U. dom  R  = 
U. Q )
1814, 17sseqtr4d 3498 . . 3  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  A  C_  U. dom  R )
19 elex 3087 . . . . . 6  |-  ( Q  e.  V  ->  Q  e.  _V )
20 uniexb 6606 . . . . . . 7  |-  ( Q  e.  _V  <->  U. Q  e. 
_V )
2120biimpi 197 . . . . . 6  |-  ( Q  e.  _V  ->  U. Q  e.  _V )
222, 19, 213syl 18 . . . . 5  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  U. Q  e.  _V )
23 ssexg 4562 . . . . 5  |-  ( ( A  C_  U. Q  /\  U. Q  e.  _V )  ->  A  e.  _V )
2414, 22, 23syl2anc 665 . . . 4  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  A  e.  _V )
25 elpwg 3984 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ~P U. dom  R  <-> 
A  C_  U. dom  R
) )
2624, 25syl 17 . . 3  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  ( A  e. 
~P U. dom  R  <->  A  C_  U. dom  R ) )
2718, 26mpbird 235 . 2  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  A  e.  ~P U.
dom  R )
28 xrltso 11429 . . . . 5  |-  <  Or  RR*
29 cnvso 5386 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
3028, 29mpbi 211 . . . 4  |-  `'  <  Or 
RR*
31 iccssxr 11706 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
32 soss 4784 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( `'  <  Or 
RR*  ->  `'  <  Or  ( 0 [,] +oo ) ) )
3331, 32ax-mp 5 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Or  ( 0 [,] +oo ) )
3430, 33mp1i 13 . . 3  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  `'  <  Or  ( 0 [,] +oo ) )
3534supexd 7964 . 2  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  )  e. 
_V )
366, 13, 27, 35fvmptd 5961 1  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  ( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   {crab 2777   _Vcvv 3078    C_ wss 3433   ~Pcpw 3976   U.cuni 4213   class class class wbr 4417    |-> cmpt 4475    Or wor 4765   `'ccnv 4844   dom cdm 4845   ran crn 4846   -->wf 5588   ` cfv 5592  (class class class)co 6296   omcom 6697    ~<_ cdom 7566   supcsup 7951   0cc0 9528   +oocpnf 9661   RR*cxr 9663    < clt 9664   [,]cicc 11627  Σ*cesum 28713  toOMeascoms 28978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-pre-lttri 9602  ax-pre-lttrn 9603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-icc 11631  df-esum 28714  df-oms 28979
This theorem is referenced by:  omsf  28983  oms0  28984  omsmon  28985  omssubaddlem  28986  omssubadd  28987
  Copyright terms: Public domain W3C validator