Step |
Hyp |
Ref |
Expression |
1 |
|
zre |
|- ( A e. ZZ -> A e. RR ) |
2 |
1
|
adantr |
|- ( ( A e. ZZ /\ M e. RR+ ) -> A e. RR ) |
3 |
|
rpre |
|- ( M e. RR+ -> M e. RR ) |
4 |
3
|
adantl |
|- ( ( A e. ZZ /\ M e. RR+ ) -> M e. RR ) |
5 |
|
rpne0 |
|- ( M e. RR+ -> M =/= 0 ) |
6 |
5
|
adantl |
|- ( ( A e. ZZ /\ M e. RR+ ) -> M =/= 0 ) |
7 |
2 4 6
|
redivcld |
|- ( ( A e. ZZ /\ M e. RR+ ) -> ( A / M ) e. RR ) |
8 |
7
|
flcld |
|- ( ( A e. ZZ /\ M e. RR+ ) -> ( |_ ` ( A / M ) ) e. ZZ ) |
9 |
8
|
3adant2 |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( |_ ` ( A / M ) ) e. ZZ ) |
10 |
|
oveq1 |
|- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) ) |
11 |
10
|
oveq1d |
|- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) |
12 |
11
|
eqeq2d |
|- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) ) |
13 |
12
|
adantl |
|- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) ) |
14 |
1
|
anim1i |
|- ( ( A e. ZZ /\ M e. RR+ ) -> ( A e. RR /\ M e. RR+ ) ) |
15 |
14
|
3adant2 |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( A e. RR /\ M e. RR+ ) ) |
16 |
|
flpmodeq |
|- ( ( A e. RR /\ M e. RR+ ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A ) |
17 |
15 16
|
syl |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A ) |
18 |
17
|
eqcomd |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) |
19 |
9 13 18
|
rspcedvd |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) ) |
20 |
|
oveq2 |
|- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) ) |
21 |
20
|
eqeq2d |
|- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) ) |
22 |
21
|
eqcoms |
|- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) ) |
23 |
22
|
rexbidv |
|- ( ( A mod M ) = B -> ( E. k e. ZZ A = ( ( k x. M ) + B ) <-> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) ) ) |
24 |
19 23
|
syl5ibrcom |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) ) |
25 |
|
oveq1 |
|- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) ) |
26 |
|
simpr |
|- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> k e. ZZ ) |
27 |
|
simpl3 |
|- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> M e. RR+ ) |
28 |
|
simpl2 |
|- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> B e. ( 0 [,) M ) ) |
29 |
|
muladdmodid |
|- ( ( k e. ZZ /\ M e. RR+ /\ B e. ( 0 [,) M ) ) -> ( ( ( k x. M ) + B ) mod M ) = B ) |
30 |
26 27 28 29
|
syl3anc |
|- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> ( ( ( k x. M ) + B ) mod M ) = B ) |
31 |
25 30
|
sylan9eqr |
|- ( ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B ) |
32 |
31
|
rexlimdva2 |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) ) |
33 |
24 32
|
impbid |
|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) ) |