Metamath Proof Explorer

Theorem ddeval0

Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018)

Ref Expression
Assertion ddeval0 
|- ( ( A C_ RR /\ -. 0 e. A ) -> ( Ddelta ` A ) = 0 )

Proof

Step Hyp Ref Expression
1 reex 
 |-  RR e. _V
2 1 ssex 
 |-  ( A C_ RR -> A e. _V )
3 elpwg 
 |-  ( A e. _V -> ( A e. ~P RR <-> A C_ RR ) )
4 3 biimpar 
 |-  ( ( A e. _V /\ A C_ RR ) -> A e. ~P RR )
5 2 4 mpancom 
 |-  ( A C_ RR -> A e. ~P RR )
6 eleq2 
 |-  ( a = A -> ( 0 e. a <-> 0 e. A ) )
7 6 ifbid 
 |-  ( a = A -> if ( 0 e. a , 1 , 0 ) = if ( 0 e. A , 1 , 0 ) )
8 df-dde 
 |-  Ddelta = ( a e. ~P RR |-> if ( 0 e. a , 1 , 0 ) )
9 1ex 
 |-  1 e. _V
10 c0ex 
 |-  0 e. _V
11 9 10 ifex 
 |-  if ( 0 e. A , 1 , 0 ) e. _V
12 7 8 11 fvmpt 
 |-  ( A e. ~P RR -> ( Ddelta ` A ) = if ( 0 e. A , 1 , 0 ) )
13 5 12 syl 
 |-  ( A C_ RR -> ( Ddelta ` A ) = if ( 0 e. A , 1 , 0 ) )
14 iffalse 
 |-  ( -. 0 e. A -> if ( 0 e. A , 1 , 0 ) = 0 )
15 13 14 sylan9eq 
 |-  ( ( A C_ RR /\ -. 0 e. A ) -> ( Ddelta ` A ) = 0 )