Metamath Proof Explorer

Theorem zdivadd

Description: Property of divisibility: if D divides A and B then it divides A + B . (Contributed by NM, 3-Oct-2008)

Ref Expression
Assertion zdivadd 
|- ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ )

Proof

Step Hyp Ref Expression
1 zcn 
 |-  ( A e. ZZ -> A e. CC )
2 zcn 
 |-  ( B e. ZZ -> B e. CC )
3 nncn 
 |-  ( D e. NN -> D e. CC )
4 nnne0 
 |-  ( D e. NN -> D =/= 0 )
5 3 4 jca 
 |-  ( D e. NN -> ( D e. CC /\ D =/= 0 ) )
6 divdir 
 |-  ( ( A e. CC /\ B e. CC /\ ( D e. CC /\ D =/= 0 ) ) -> ( ( A + B ) / D ) = ( ( A / D ) + ( B / D ) ) )
7 1 2 5 6 syl3an 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ D e. NN ) -> ( ( A + B ) / D ) = ( ( A / D ) + ( B / D ) ) )
8 7 3comr 
 |-  ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) -> ( ( A + B ) / D ) = ( ( A / D ) + ( B / D ) ) )
9 8 adantr 
 |-  ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) = ( ( A / D ) + ( B / D ) ) )
10 zaddcl 
 |-  ( ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) -> ( ( A / D ) + ( B / D ) ) e. ZZ )
11 10 adantl 
 |-  ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A / D ) + ( B / D ) ) e. ZZ )
12 9 11 eqeltrd 
 |-  ( ( ( D e. NN /\ A e. ZZ /\ B e. ZZ ) /\ ( ( A / D ) e. ZZ /\ ( B / D ) e. ZZ ) ) -> ( ( A + B ) / D ) e. ZZ )