Metamath Proof Explorer


Theorem adddirp1d

Description: Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses adddirp1d.a
|- ( ph -> A e. CC )
adddirp1d.b
|- ( ph -> B e. CC )
Assertion adddirp1d
|- ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )

Proof

Step Hyp Ref Expression
1 adddirp1d.a
 |-  ( ph -> A e. CC )
2 adddirp1d.b
 |-  ( ph -> B e. CC )
3 1cnd
 |-  ( ph -> 1 e. CC )
4 1 3 2 adddird
 |-  ( 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 eqtrd
 |-  ( ph -> ( ( A + 1 ) x. B ) = ( ( A x. B ) + B ) )