Metamath Proof Explorer

Theorem neglcm

Description: Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020)

Ref Expression
Assertion neglcm 
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )

Proof

Step Hyp Ref Expression
1 lcmneg 
 |-  ( ( N e. ZZ /\ M e. ZZ ) -> ( N lcm -u M ) = ( N lcm M ) )
2 1 ancoms 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( N lcm -u M ) = ( N lcm M ) )
3 znegcl 
 |-  ( M e. ZZ -> -u M e. ZZ )
4 lcmcom 
 |-  ( ( -u M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( N lcm -u M ) )
5 3 4 sylan 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( N lcm -u M ) )
6 lcmcom 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( M lcm N ) = ( N lcm M ) )
7 2 5 6 3eqtr4d 
 |-  ( ( M e. ZZ /\ N e. ZZ ) -> ( -u M lcm N ) = ( M lcm N ) )