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

Theorem carsgval 29208
Description: Value of the Caratheodory sigma-Algebra construction function. (Contributed by Thierry Arnoux, 17-May-2020.)
Hypotheses
Ref Expression
carsgval.1  |-  ( ph  ->  O  e.  V )
carsgval.2  |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )
Assertion
Ref Expression
carsgval  |-  ( ph  ->  (toCaraSiga `  M )  =  { a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `  ( e  i^i  a ) ) +e ( M `
 ( e  \ 
a ) ) )  =  ( M `  e ) } )
Distinct variable groups:    M, a,
e    O, a, e    ph, a,
e
Allowed substitution hints:    V( e, a)

Proof of Theorem carsgval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 df-carsg 29207 . . 3  |- toCaraSiga  =  ( m  e.  _V  |->  { a  e.  ~P U. dom  m  |  A. e  e.  ~P  U. dom  m
( ( m `  ( e  i^i  a
) ) +e
( m `  (
e  \  a )
) )  =  ( m `  e ) } )
21a1i 11 . 2  |-  ( ph  -> toCaraSiga  =  ( m  e. 
_V  |->  { a  e. 
~P U. dom  m  | 
A. e  e.  ~P  U.
dom  m ( ( m `  ( e  i^i  a ) ) +e ( m `
 ( e  \ 
a ) ) )  =  ( m `  e ) } ) )
3 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  m  =  M )  ->  m  =  M )
43dmeqd 5042 . . . . . . 7  |-  ( (
ph  /\  m  =  M )  ->  dom  m  =  dom  M )
5 carsgval.2 . . . . . . . . 9  |-  ( ph  ->  M : ~P O --> ( 0 [,] +oo ) )
6 fdm 5745 . . . . . . . . 9  |-  ( M : ~P O --> ( 0 [,] +oo )  ->  dom  M  =  ~P O
)
75, 6syl 17 . . . . . . . 8  |-  ( ph  ->  dom  M  =  ~P O )
87adantr 472 . . . . . . 7  |-  ( (
ph  /\  m  =  M )  ->  dom  M  =  ~P O )
94, 8eqtrd 2505 . . . . . 6  |-  ( (
ph  /\  m  =  M )  ->  dom  m  =  ~P O
)
109unieqd 4200 . . . . 5  |-  ( (
ph  /\  m  =  M )  ->  U. dom  m  =  U. ~P O
)
11 unipw 4650 . . . . 5  |-  U. ~P O  =  O
1210, 11syl6eq 2521 . . . 4  |-  ( (
ph  /\  m  =  M )  ->  U. dom  m  =  O )
1312pweqd 3947 . . 3  |-  ( (
ph  /\  m  =  M )  ->  ~P U.
dom  m  =  ~P O )
14 fveq1 5878 . . . . . . 7  |-  ( m  =  M  ->  (
m `  ( e  i^i  a ) )  =  ( M `  (
e  i^i  a )
) )
15 fveq1 5878 . . . . . . 7  |-  ( m  =  M  ->  (
m `  ( e  \  a ) )  =  ( M `  ( e  \  a
) ) )
1614, 15oveq12d 6326 . . . . . 6  |-  ( m  =  M  ->  (
( m `  (
e  i^i  a )
) +e ( m `  ( e 
\  a ) ) )  =  ( ( M `  ( e  i^i  a ) ) +e ( M `
 ( e  \ 
a ) ) ) )
17 fveq1 5878 . . . . . 6  |-  ( m  =  M  ->  (
m `  e )  =  ( M `  e ) )
1816, 17eqeq12d 2486 . . . . 5  |-  ( m  =  M  ->  (
( ( m `  ( e  i^i  a
) ) +e
( m `  (
e  \  a )
) )  =  ( m `  e )  <-> 
( ( M `  ( e  i^i  a
) ) +e
( M `  (
e  \  a )
) )  =  ( M `  e ) ) )
1918adantl 473 . . . 4  |-  ( (
ph  /\  m  =  M )  ->  (
( ( m `  ( e  i^i  a
) ) +e
( m `  (
e  \  a )
) )  =  ( m `  e )  <-> 
( ( M `  ( e  i^i  a
) ) +e
( M `  (
e  \  a )
) )  =  ( M `  e ) ) )
2013, 19raleqbidv 2987 . . 3  |-  ( (
ph  /\  m  =  M )  ->  ( A. e  e.  ~P  U.
dom  m ( ( m `  ( e  i^i  a ) ) +e ( m `
 ( e  \ 
a ) ) )  =  ( m `  e )  <->  A. e  e.  ~P  O ( ( M `  ( e  i^i  a ) ) +e ( M `
 ( e  \ 
a ) ) )  =  ( M `  e ) ) )
2113, 20rabeqbidv 3026 . 2  |-  ( (
ph  /\  m  =  M )  ->  { a  e.  ~P U. dom  m  |  A. e  e.  ~P  U. dom  m
( ( m `  ( e  i^i  a
) ) +e
( m `  (
e  \  a )
) )  =  ( m `  e ) }  =  { a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `
 ( e  i^i  a ) ) +e ( M `  ( e  \  a
) ) )  =  ( M `  e
) } )
22 carsgval.1 . . . 4  |-  ( ph  ->  O  e.  V )
23 pwexg 4585 . . . 4  |-  ( O  e.  V  ->  ~P O  e.  _V )
2422, 23syl 17 . . 3  |-  ( ph  ->  ~P O  e.  _V )
25 fex 6155 . . 3  |-  ( ( M : ~P O --> ( 0 [,] +oo )  /\  ~P O  e. 
_V )  ->  M  e.  _V )
265, 24, 25syl2anc 673 . 2  |-  ( ph  ->  M  e.  _V )
27 rabexg 4549 . . 3  |-  ( ~P O  e.  _V  ->  { a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `
 ( e  i^i  a ) ) +e ( M `  ( e  \  a
) ) )  =  ( M `  e
) }  e.  _V )
2822, 23, 273syl 18 . 2  |-  ( ph  ->  { a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `  ( e  i^i  a ) ) +e ( M `
 ( e  \ 
a ) ) )  =  ( M `  e ) }  e.  _V )
292, 21, 26, 28fvmptd 5969 1  |-  ( ph  ->  (toCaraSiga `  M )  =  { a  e.  ~P O  |  A. e  e.  ~P  O ( ( M `  ( e  i^i  a ) ) +e ( M `
 ( e  \ 
a ) ) )  =  ( M `  e ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031    \ cdif 3387    i^i cin 3389   ~Pcpw 3942   U.cuni 4190    |-> cmpt 4454   dom cdm 4839   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   +oocpnf 9690   +ecxad 11430   [,]cicc 11663  toCaraSigaccarsg 29206
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-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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  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-ral 2761  df-rex 2762  df-reu 2763  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-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  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-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-carsg 29207
This theorem is referenced by:  carsgcl  29209  elcarsg  29210
  Copyright terms: Public domain W3C validator