Metamath Proof Explorer

Theorem acongneg2

Description: Negate right side of alternating congruence. Makes essential use of the "alternating" part. (Contributed by Stefan O'Rear, 3-Oct-2014)

Ref Expression
Assertion acongneg2 
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - C ) \/ A || ( B - -u C ) ) )

Proof

Step Hyp Ref Expression
1 zcn 
 |-  ( C e. ZZ -> C e. CC )
2 1 3ad2ant3 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> C e. CC )
3 2 negnegd 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> -u -u C = C )
4 3 oveq2d 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( B - -u -u C ) = ( B - C ) )
5 4 breq2d 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( B - -u -u C ) <-> A || ( B - C ) ) )
6 5 biimpd 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( B - -u -u C ) -> A || ( B - C ) ) )
7 6 orim2d 
 |-  ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) -> ( A || ( B - -u C ) \/ A || ( B - C ) ) ) )
8 7 imp 
 |-  ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - -u C ) \/ A || ( B - C ) ) )
9 8 orcomd 
 |-  ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - C ) \/ A || ( B - -u C ) ) )