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Metamath Proof Explorer

Theorem modge0

Description: The modulo operation is nonnegative. (Contributed by NM, 10-Nov-2008)

Ref Expression
Assertion modge0 
|- ( ( A e. RR /\ B e. RR+ ) -> 0 <_ ( A mod B ) )

Proof

Step Hyp Ref Expression
1 fldivle 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
2 refldivcl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. RR )
3 simpl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> A e. RR )
4 rpregt0 
 |-  ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
5 4 adantl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( B e. RR /\ 0 < B ) )
6 lemuldiv2 
 |-  ( ( ( |_ ` ( A / B ) ) e. RR /\ A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( B x. ( |_ ` ( A / B ) ) ) <_ A <-> ( |_ ` ( A / B ) ) <_ ( A / B ) ) )
7 2 3 5 6 syl3anc 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( ( B x. ( |_ ` ( A / B ) ) ) <_ A <-> ( |_ ` ( A / B ) ) <_ ( A / B ) ) )
8 1 7 mpbird 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) <_ A )
9 rpre 
 |-  ( B e. RR+ -> B e. RR )
10 9 adantl 
 |-  ( ( A e. RR /\ B e. RR+ ) -> B e. RR )
11 10 2 remulcld 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. RR )
12 subge0 
 |-  ( ( A e. RR /\ ( B x. ( |_ ` ( A / B ) ) ) e. RR ) -> ( 0 <_ ( A - ( B x. ( |_ ` ( A / B ) ) ) ) <-> ( B x. ( |_ ` ( A / B ) ) ) <_ A ) )
13 11 12 syldan 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( 0 <_ ( A - ( B x. ( |_ ` ( A / B ) ) ) ) <-> ( B x. ( |_ ` ( A / B ) ) ) <_ A ) )
14 8 13 mpbird 
 |-  ( ( A e. RR /\ B e. RR+ ) -> 0 <_ ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
15 modval 
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) )
16 14 15 breqtrrd 
 |-  ( ( A e. RR /\ B e. RR+ ) -> 0 <_ ( A mod B ) )