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Metamath Proof Explorer

Theorem qnegcl

Description: Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004) (Revised by Mario Carneiro, 15-Sep-2014)

Ref Expression
Assertion qnegcl 
|- ( A e. QQ -> -u A e. QQ )

Proof

Step Hyp Ref Expression
1 elq 
 |-  ( A e. QQ <-> E. x e. ZZ E. y e. NN A = ( x / y ) )
2 zcn 
 |-  ( x e. ZZ -> x e. CC )
3 2 adantr 
 |-  ( ( x e. ZZ /\ y e. NN ) -> x e. CC )
4 nncn 
 |-  ( y e. NN -> y e. CC )
5 4 adantl 
 |-  ( ( x e. ZZ /\ y e. NN ) -> y e. CC )
6 nnne0 
 |-  ( y e. NN -> y =/= 0 )
7 6 adantl 
 |-  ( ( x e. ZZ /\ y e. NN ) -> y =/= 0 )
8 3 5 7 divnegd 
 |-  ( ( x e. ZZ /\ y e. NN ) -> -u ( x / y ) = ( -u x / y ) )
9 znegcl 
 |-  ( x e. ZZ -> -u x e. ZZ )
10 znq 
 |-  ( ( -u x e. ZZ /\ y e. NN ) -> ( -u x / y ) e. QQ )
11 9 10 sylan 
 |-  ( ( x e. ZZ /\ y e. NN ) -> ( -u x / y ) e. QQ )
12 8 11 eqeltrd 
 |-  ( ( x e. ZZ /\ y e. NN ) -> -u ( x / y ) e. QQ )
13 negeq 
 |-  ( A = ( x / y ) -> -u A = -u ( x / y ) )
14 13 eleq1d 
 |-  ( A = ( x / y ) -> ( -u A e. QQ <-> -u ( x / y ) e. QQ ) )
15 12 14 syl5ibrcom 
 |-  ( ( x e. ZZ /\ y e. NN ) -> ( A = ( x / y ) -> -u A e. QQ ) )
16 15 rexlimivv 
 |-  ( E. x e. ZZ E. y e. NN A = ( x / y ) -> -u A e. QQ )
17 1 16 sylbi 
 |-  ( A e. QQ -> -u A e. QQ )