Metamath Proof Explorer

Theorem dmdcan

Description: Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013) (Proof shortened by Fan Zheng, 3-Jul-2016)

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

Proof

Step Hyp Ref Expression
1 simp1l 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> A e. CC )
2 simp3 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> C e. CC )
3 simp1r 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> A =/= 0 )
4 divcl 
 |-  ( ( C e. CC /\ A e. CC /\ A =/= 0 ) -> ( C / A ) e. CC )
5 2 1 3 4 syl3anc 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( C / A ) e. CC )
6 simp2l 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> B e. CC )
7 simp2r 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> B =/= 0 )
8 div23 
 |-  ( ( A e. CC /\ ( C / A ) e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A x. ( C / A ) ) / B ) = ( ( A / B ) x. ( C / A ) ) )
9 1 5 6 7 8 syl112anc 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A x. ( C / A ) ) / B ) = ( ( A / B ) x. ( C / A ) ) )
10 divcan2 
 |-  ( ( C e. CC /\ A e. CC /\ A =/= 0 ) -> ( A x. ( C / A ) ) = C )
11 2 1 3 10 syl3anc 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( A x. ( C / A ) ) = C )
12 11 oveq1d 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A x. ( C / A ) ) / B ) = ( C / B ) )
13 9 12 eqtr3d 
 |-  ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) /\ C e. CC ) -> ( ( A / B ) x. ( C / A ) ) = ( C / B ) )