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Theorem omsfval 27933
Description: Value of the outer measure evaluated for a given set 
A. (Contributed by Thierry Arnoux, 15-Sep-2019.)
Assertion
Ref Expression
omsfval  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  -> 
( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P ~P U. dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
Distinct variable groups:    x, R, y, z    x, A, y, z    z, O
Allowed substitution hints:    O( x, y)

Proof of Theorem omsfval
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 simp2 997 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  R : ~P O --> ( 0 [,] +oo ) )
2 pwexg 4631 . . . . 5  |-  ( O  e.  _V  ->  ~P O  e.  _V )
323ad2ant1 1017 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  ~P O  e.  _V )
4 fex 6133 . . . 4  |-  ( ( R : ~P O --> ( 0 [,] +oo )  /\  ~P O  e. 
_V )  ->  R  e.  _V )
51, 3, 4syl2anc 661 . . 3  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  R  e.  _V )
6 omsval 27932 . . 3  |-  ( R  e.  _V  ->  (toOMeas `  R )  =  ( a  e.  ~P U. dom  R  |->  sup ( ran  (
x  e.  { z  e.  ~P ~P U. dom  R  |  ( a 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
75, 6syl 16 . 2  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  -> 
(toOMeas `  R )  =  ( a  e.  ~P U.
dom  R  |->  sup ( ran  ( x  e.  {
z  e.  ~P ~P U.
dom  R  |  (
a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
8 simpr 461 . . . . . . . 8  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  a  =  A )
98sseq1d 3531 . . . . . . 7  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  (
a  C_  U. z  <->  A 
C_  U. z ) )
109anbi1d 704 . . . . . 6  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  (
( a  C_  U. z  /\  z  ~<_  om )  <->  ( A  C_  U. z  /\  z  ~<_  om )
) )
1110rabbidv 3105 . . . . 5  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  { z  e.  ~P ~P U. dom  R  |  ( a 
C_  U. z  /\  z  ~<_  om ) }  =  {
z  e.  ~P ~P U.
dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } )
1211mpteq1d 4528 . . . 4  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  (
x  e.  { z  e.  ~P ~P U. dom  R  |  ( a 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) )  =  ( x  e.  { z  e.  ~P ~P U. dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) )
1312rneqd 5230 . . 3  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  ran  ( x  e.  { z  e.  ~P ~P U. dom  R  |  ( a 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) )  =  ran  ( x  e.  { z  e.  ~P ~P U. dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) )
1413supeq1d 7906 . 2  |-  ( ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O
)  /\  a  =  A )  ->  sup ( ran  ( x  e. 
{ z  e.  ~P ~P U. dom  R  | 
( a  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  )  =  sup ( ran  ( x  e. 
{ z  e.  ~P ~P U. dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x ( R `  y ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
15 simp3 998 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  C_  O )
16 fdm 5735 . . . . . . 7  |-  ( R : ~P O --> ( 0 [,] +oo )  ->  dom  R  =  ~P O
)
171, 16syl 16 . . . . . 6  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  dom  R  =  ~P O
)
1817unieqd 4255 . . . . 5  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  U. dom  R  =  U. ~P O )
19 unipw 4697 . . . . 5  |-  U. ~P O  =  O
2018, 19syl6req 2525 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  O  =  U. dom  R
)
2115, 20sseqtrd 3540 . . 3  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  C_  U. dom  R
)
22 simp1 996 . . . . 5  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  O  e.  _V )
23 ssexg 4593 . . . . 5  |-  ( ( A  C_  O  /\  O  e.  _V )  ->  A  e.  _V )
2415, 22, 23syl2anc 661 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  e.  _V )
25 elpwg 4018 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ~P U. dom  R  <-> 
A  C_  U. dom  R
) )
2624, 25syl 16 . . 3  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  -> 
( A  e.  ~P U.
dom  R  <->  A  C_  U. dom  R ) )
2721, 26mpbird 232 . 2  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  e.  ~P U. dom  R )
28 xrltso 11347 . . . . 5  |-  <  Or  RR*
29 cnvso 5546 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
3028, 29mpbi 208 . . . 4  |-  `'  <  Or 
RR*
31 iccssxr 11607 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
32 soss 4818 . . . . 5  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  ( `'  <  Or 
RR*  ->  `'  <  Or  ( 0 [,] +oo ) ) )
3331, 32ax-mp 5 . . . 4  |-  ( `'  <  Or  RR*  ->  `'  <  Or  ( 0 [,] +oo ) )
3430, 33mp1i 12 . . 3  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  `'  <  Or  ( 0 [,] +oo ) )
3534supexd 7913 . 2  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  sup ( ran  ( x  e.  { z  e. 
~P ~P U. dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* y  e.  x
( R `  y
) ) ,  ( 0 [,] +oo ) ,  `'  <  )  e. 
_V )
367, 14, 27, 35fvmptd 5955 1  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  -> 
( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P ~P U. 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   `'ccnv 4998   dom cdm 4999   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284   omcom 6684    ~<_ cdom 7514   supcsup 7900   0cc0 9492   +oocpnf 9625   RR*cxr 9627    < clt 9628   [,]cicc 11532  Σ*cesum 27708  toOMeascoms 27930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-icc 11536  df-esum 27709  df-oms 27931
This theorem is referenced by:  oms0  27934  omsmon  27935
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