Metamath Proof Explorer

Theorem div23

Description: A commutative/associative law for division. (Contributed by NM, 2-Aug-2004)

Ref Expression
Assertion div23 
|- ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) )

Proof

Step Hyp Ref Expression
1 mulcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2 1 oveq1d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) / C ) = ( ( B x. A ) / C ) )
3 2 3adant3 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A x. B ) / C ) = ( ( B x. A ) / C ) )
4 divass 
 |-  ( ( B e. CC /\ A e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( B x. A ) / C ) = ( B x. ( A / C ) ) )
5 4 3com12 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( B x. A ) / C ) = ( B x. ( A / C ) ) )
6 simp2 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> B e. CC )
7 divcl 
 |-  ( ( A e. CC /\ C e. CC /\ C =/= 0 ) -> ( A / C ) e. CC )
8 7 3expb 
 |-  ( ( A e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / C ) e. CC )
9 8 3adant2 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( A / C ) e. CC )
10 6 9 mulcomd 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( B x. ( A / C ) ) = ( ( A / C ) x. B ) )
11 3 5 10 3eqtrd 
 |-  ( ( A e. CC /\ B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( A x. B ) / C ) = ( ( A / C ) x. B ) )