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Metamath Proof Explorer

Theorem div13

Description: A commutative/associative law for division. (Contributed by NM, 22-Apr-2005)

Ref Expression
Assertion div13 
|- ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )

Proof

Step Hyp Ref Expression
1 mulcom 
 |-  ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
2 1 oveq1d 
 |-  ( ( A e. CC /\ C e. CC ) -> ( ( A x. C ) / B ) = ( ( C x. A ) / B ) )
3 2 3adant2 
 |-  ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A x. C ) / B ) = ( ( C x. A ) / B ) )
4 div23 
 |-  ( ( A e. CC /\ C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A x. C ) / B ) = ( ( A / B ) x. C ) )
5 4 3com23 
 |-  ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A x. C ) / B ) = ( ( A / B ) x. C ) )
6 div23 
 |-  ( ( C e. CC /\ A e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( C x. A ) / B ) = ( ( C / B ) x. A ) )
7 6 3coml 
 |-  ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( C x. A ) / B ) = ( ( C / B ) x. A ) )
8 3 5 7 3eqtr3d 
 |-  ( ( A e. CC /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A / B ) x. C ) = ( ( C / B ) x. A ) )