Metamath Proof Explorer

Theorem elcarsg

Description: Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020)

Ref Expression
Hypotheses carsgval.1 
|- ( ph -> O e. V )
carsgval.2 
|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
Assertion elcarsg 
|- ( ph -> ( A e. ( toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )

Proof

Step Hyp Ref Expression
1 carsgval.1 
 |-  ( ph -> O e. V )
2 carsgval.2 
 |-  ( ph -> M : ~P O --> ( 0 [,] +oo ) )
3 1 2 carsgval 
 |-  ( ph -> ( toCaraSiga ` M ) = { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } )
4 3 eleq2d 
 |-  ( ph -> ( A e. ( toCaraSiga ` M ) <-> A e. { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } ) )
5 ineq2 
 |-  ( a = A -> ( e i^i a ) = ( e i^i A ) )
6 5 fveq2d 
 |-  ( a = A -> ( M ` ( e i^i a ) ) = ( M ` ( e i^i A ) ) )
7 difeq2 
 |-  ( a = A -> ( e \ a ) = ( e \ A ) )
8 7 fveq2d 
 |-  ( a = A -> ( M ` ( e \ a ) ) = ( M ` ( e \ A ) ) )
9 6 8 oveq12d 
 |-  ( a = A -> ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) )
10 9 eqeq1d 
 |-  ( a = A -> ( ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) <-> ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) )
11 10 ralbidv 
 |-  ( a = A -> ( A. e e. ~P O ( ( 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 ) ) )
12 11 elrab 
 |-  ( A e. { a e. ~P O | A. e e. ~P O ( ( 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 ) ) )
13 elex 
 |-  ( A e. ~P O -> A e. _V )
14 13 a1i 
 |-  ( ph -> ( A e. ~P O -> A e. _V ) )
15 1 adantr 
 |-  ( ( ph /\ A C_ O ) -> O e. V )
16 simpr 
 |-  ( ( ph /\ A C_ O ) -> A C_ O )
17 15 16 ssexd 
 |-  ( ( ph /\ A C_ O ) -> A e. _V )
18 17 ex 
 |-  ( ph -> ( A C_ O -> A e. _V ) )
19 elpwg 
 |-  ( A e. _V -> ( A e. ~P O <-> A C_ O ) )
20 19 a1i 
 |-  ( ph -> ( A e. _V -> ( A e. ~P O <-> A C_ O ) ) )
21 14 18 20 pm5.21ndd 
 |-  ( ph -> ( A e. ~P O <-> A C_ O ) )
22 21 anbi1d 
 |-  ( ph -> ( ( A e. ~P O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
23 12 22 syl5bb 
 |-  ( ph -> ( A e. { a e. ~P O | A. e e. ~P O ( ( M ` ( e i^i a ) ) +e ( M ` ( e \ a ) ) ) = ( M ` e ) } <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
24 4 23 bitrd 
 |-  ( ph -> ( A e. ( toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )