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 ) ) |