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Metamath Proof Explorer

Theorem modmulmod

Description: The product of a real number modulo a positive real number and an integer equals the product of the real number and the integer modulo the positive real number. (Contributed by Alexander van der Vekens, 17-May-2018)

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

Proof

Step Hyp Ref Expression
1 modcl 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) e. RR )
2 simpl 
 |-  ( ( A e. RR /\ M e. RR+ ) -> A e. RR )
3 1 2 jca 
 |-  ( ( A e. RR /\ M e. RR+ ) -> ( ( A mod M ) e. RR /\ A e. RR ) )
4 3 3adant2 
 |-  ( ( A e. RR /\ B e. ZZ /\ M e. RR+ ) -> ( ( A mod M ) e. RR /\ A e. RR ) )
5 3simpc 
 |-  ( ( A e. RR /\ B e. ZZ /\ M e. RR+ ) -> ( B e. ZZ /\ 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. ZZ /\ M e. RR+ ) -> ( ( A mod M ) mod M ) = ( A mod M ) )
8 modmul1 
 |-  ( ( ( ( A mod M ) e. RR /\ A e. RR ) /\ ( B e. ZZ /\ M e. RR+ ) /\ ( ( A mod M ) mod M ) = ( A mod M ) ) -> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) )
9 4 5 7 8 syl3anc 
 |-  ( ( A e. RR /\ B e. ZZ /\ M e. RR+ ) -> ( ( ( A mod M ) x. B ) mod M ) = ( ( A x. B ) mod M ) )