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Theorem omsfval 26714
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 989 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  R : ~P O --> ( 0 [,] +oo ) )
2 pwexg 4481 . . . . 5  |-  ( O  e.  _V  ->  ~P O  e.  _V )
323ad2ant1 1009 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  ~P O  e.  _V )
4 fex 5955 . . . 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 26713 . . 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 3388 . . . . . . 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 2969 . . . . 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 4378 . . . 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 5072 . . 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 7701 . 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 990 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  C_  O )
16 fdm 5568 . . . . . . 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 4106 . . . . 5  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  U. dom  R  =  U. ~P O )
19 unipw 4547 . . . . 5  |-  U. ~P O  =  O
2018, 19syl6req 2492 . . . 4  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  O  =  U. dom  R
)
2115, 20sseqtrd 3397 . . 3  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  A  C_  U. dom  R
)
22 simp1 988 . . . . 5  |-  ( ( O  e.  _V  /\  R : ~P O --> ( 0 [,] +oo )  /\  A  C_  O )  ->  O  e.  _V )
23 ssexg 4443 . . . . 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 3873 . . . 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 11123 . . . . 5  |-  <  Or  RR*
29 cnvso 5381 . . . . 5  |-  (  < 
Or  RR*  <->  `'  <  Or  RR* )
3028, 29mpbi 208 . . . 4  |-  `'  <  Or 
RR*
31 iccssxr 11383 . . . . 5  |-  ( 0 [,] +oo )  C_  RR*
32 soss 4664 . . . . 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 7708 . 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 5784 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 965    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297    e. cmpt 4355    Or wor 4645   `'ccnv 4844   dom cdm 4845   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096   omcom 6481    ~<_ cdom 7313   supcsup 7695   0cc0 9287   +oocpnf 9420   RR*cxr 9422    < clt 9423   [,]cicc 11308  Σ*cesum 26488  toOMeascoms 26711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-icc 11312  df-esum 26489  df-oms 26712
This theorem is referenced by:  oms0  26715  omsmon  26716
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