Metamath Proof Explorer


Theorem mulsubfacd

Description: Multiplication followed by the subtraction of a factor. (Contributed by Alexander van der Vekens, 28-Aug-2018)

Ref Expression
Hypotheses muls1d.1
|- ( ph -> A e. CC )
muls1d.2
|- ( ph -> B e. CC )
Assertion mulsubfacd
|- ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) )

Proof

Step Hyp Ref Expression
1 muls1d.1
 |-  ( ph -> A e. CC )
2 muls1d.2
 |-  ( ph -> B e. CC )
3 1cnd
 |-  ( ph -> 1 e. CC )
4 1 3 2 subdird
 |-  ( ph -> ( ( A - 1 ) x. B ) = ( ( A x. B ) - ( 1 x. B ) ) )
5 2 mulid2d
 |-  ( ph -> ( 1 x. B ) = B )
6 5 oveq2d
 |-  ( ph -> ( ( A x. B ) - ( 1 x. B ) ) = ( ( A x. B ) - B ) )
7 4 6 eqtr2d
 |-  ( ph -> ( ( A x. B ) - B ) = ( ( A - 1 ) x. B ) )