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Metamath Proof Explorer

Theorem mbfneg

Description: The negative of a measurable function is measurable. (Contributed by Mario Carneiro, 31-Jul-2014)

Ref Expression
Hypotheses mbfneg.1 
|- ( ( ph /\ x e. A ) -> B e. V )
mbfneg.2 
|- ( ph -> ( x e. A |-> B ) e. MblFn )
Assertion mbfneg 
|- ( ph -> ( x e. A |-> -u B ) e. MblFn )

Proof

Step Hyp Ref Expression
1 mbfneg.1 
 |-  ( ( ph /\ x e. A ) -> B e. V )
2 mbfneg.2 
 |-  ( ph -> ( x e. A |-> B ) e. MblFn )
3 eqid 
 |-  ( x e. A |-> B ) = ( x e. A |-> B )
4 3 1 dmmptd 
 |-  ( ph -> dom ( x e. A |-> B ) = A )
5 2 dmexd 
 |-  ( ph -> dom ( x e. A |-> B ) e. _V )
6 4 5 eqeltrrd 
 |-  ( ph -> A e. _V )
7 neg1rr 
 |-  -u 1 e. RR
8 7 a1i 
 |-  ( ( ph /\ x e. A ) -> -u 1 e. RR )
9 fconstmpt 
 |-  ( A X. { -u 1 } ) = ( x e. A |-> -u 1 )
10 9 a1i 
 |-  ( ph -> ( A X. { -u 1 } ) = ( x e. A |-> -u 1 ) )
11 eqidd 
 |-  ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
12 6 8 1 10 11 offval2 
 |-  ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) = ( x e. A |-> ( -u 1 x. B ) ) )
13 2 1 mbfmptcl 
 |-  ( ( ph /\ x e. A ) -> B e. CC )
14 13 mulm1d 
 |-  ( ( ph /\ x e. A ) -> ( -u 1 x. B ) = -u B )
15 14 mpteq2dva 
 |-  ( ph -> ( x e. A |-> ( -u 1 x. B ) ) = ( x e. A |-> -u B ) )
16 12 15 eqtrd 
 |-  ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) = ( x e. A |-> -u B ) )
17 7 a1i 
 |-  ( ph -> -u 1 e. RR )
18 13 fmpttd 
 |-  ( ph -> ( x e. A |-> B ) : A --> CC )
19 2 17 18 mbfmulc2re 
 |-  ( ph -> ( ( A X. { -u 1 } ) oF x. ( x e. A |-> B ) ) e. MblFn )
20 16 19 eqeltrrd 
 |-  ( ph -> ( x e. A |-> -u B ) e. MblFn )