# 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 ) )`