Metamath Proof Explorer


Theorem negsubdi3d

Description: Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses negsubdi3d.1
|- ( ph -> A e. CC )
negsubdi3d.2
|- ( ph -> B e. CC )
Assertion negsubdi3d
|- ( ph -> -u ( A - B ) = ( -u A - -u B ) )

Proof

Step Hyp Ref Expression
1 negsubdi3d.1
 |-  ( ph -> A e. CC )
2 negsubdi3d.2
 |-  ( ph -> B e. CC )
3 1 2 negsubdi2d
 |-  ( ph -> -u ( A - B ) = ( B - A ) )
4 1 2 neg2subd
 |-  ( ph -> ( -u A - -u B ) = ( B - A ) )
5 3 4 eqtr4d
 |-  ( ph -> -u ( A - B ) = ( -u A - -u B ) )