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 ) )