Theorem modsubmod

Description: The difference of a real number modulo a positive real number and another real number equals the difference of the two real numbers modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018)

	Ref	Expression
Assertion	modsubmod	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) )

Proof

Step	Hyp	Ref	Expression
1		modcl	|- ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) e. RR )
2	1	3adant2	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( A mod M ) e. RR )
3		simp1	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> A e. RR )
4		simp2	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> B e. RR )
5		simp3	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> M e. RR+ )
6		modabs2	|- ( ( A e. RR /\ M e. RR+ ) -> ( ( A mod M ) mod M ) = ( A mod M ) )
7	6	3adant2	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( A mod M ) mod M ) = ( A mod M ) )
8		eqidd	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( B mod M ) = ( B mod M ) )
9	2 3 4 4 5 7 8	modsub12d	|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) - B ) mod M ) = ( ( A - B ) mod M ) )