Step |
Hyp |
Ref |
Expression |
1 |
|
modvalr |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( ( |_ ` ( A / B ) ) x. B ) ) ) |
2 |
1
|
eqcomd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A - ( ( |_ ` ( A / B ) ) x. B ) ) = ( A mod B ) ) |
3 |
|
recn |
|- ( A e. RR -> A e. CC ) |
4 |
3
|
adantr |
|- ( ( A e. RR /\ B e. RR+ ) -> A e. CC ) |
5 |
|
rerpdivcl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR ) |
6 |
|
flcl |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. ZZ ) |
7 |
6
|
zcnd |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. CC ) |
8 |
5 7
|
syl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. CC ) |
9 |
|
rpcn |
|- ( B e. RR+ -> B e. CC ) |
10 |
9
|
adantl |
|- ( ( A e. RR /\ B e. RR+ ) -> B e. CC ) |
11 |
8 10
|
mulcld |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( |_ ` ( A / B ) ) x. B ) e. CC ) |
12 |
|
modcl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) e. RR ) |
13 |
12
|
recnd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) e. CC ) |
14 |
4 11 13
|
subaddd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A - ( ( |_ ` ( A / B ) ) x. B ) ) = ( A mod B ) <-> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A ) ) |
15 |
2 14
|
mpbid |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( ( |_ ` ( A / B ) ) x. B ) + ( A mod B ) ) = A ) |