Step |
Hyp |
Ref |
Expression |
1 |
|
modval |
|- ( ( A e. RR /\ D e. RR+ ) -> ( A mod D ) = ( A - ( D x. ( |_ ` ( A / D ) ) ) ) ) |
2 |
|
modval |
|- ( ( B e. RR /\ D e. RR+ ) -> ( B mod D ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) |
3 |
1 2
|
eqeqan12d |
|- ( ( ( A e. RR /\ D e. RR+ ) /\ ( B e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) ) |
4 |
3
|
anandirs |
|- ( ( ( A e. RR /\ B e. RR ) /\ D e. RR+ ) -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) ) |
5 |
4
|
adantrl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) <-> ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) ) ) |
6 |
|
oveq1 |
|- ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) = ( B - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) |
7 |
5 6
|
syl6bi |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) ) |
8 |
|
recn |
|- ( A e. RR -> A e. CC ) |
9 |
8
|
adantr |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> A e. CC ) |
10 |
|
recn |
|- ( C e. RR -> C e. CC ) |
11 |
10
|
ad2antrl |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> C e. CC ) |
12 |
|
rpcn |
|- ( D e. RR+ -> D e. CC ) |
13 |
12
|
adantl |
|- ( ( A e. RR /\ D e. RR+ ) -> D e. CC ) |
14 |
|
rerpdivcl |
|- ( ( A e. RR /\ D e. RR+ ) -> ( A / D ) e. RR ) |
15 |
|
reflcl |
|- ( ( A / D ) e. RR -> ( |_ ` ( A / D ) ) e. RR ) |
16 |
15
|
recnd |
|- ( ( A / D ) e. RR -> ( |_ ` ( A / D ) ) e. CC ) |
17 |
14 16
|
syl |
|- ( ( A e. RR /\ D e. RR+ ) -> ( |_ ` ( A / D ) ) e. CC ) |
18 |
13 17
|
mulcld |
|- ( ( A e. RR /\ D e. RR+ ) -> ( D x. ( |_ ` ( A / D ) ) ) e. CC ) |
19 |
18
|
adantrl |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( D x. ( |_ ` ( A / D ) ) ) e. CC ) |
20 |
9 11 19
|
addsubd |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) ) |
21 |
20
|
adantlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) ) |
22 |
|
recn |
|- ( B e. RR -> B e. CC ) |
23 |
22
|
adantr |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> B e. CC ) |
24 |
10
|
ad2antrl |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> C e. CC ) |
25 |
12
|
adantl |
|- ( ( B e. RR /\ D e. RR+ ) -> D e. CC ) |
26 |
|
rerpdivcl |
|- ( ( B e. RR /\ D e. RR+ ) -> ( B / D ) e. RR ) |
27 |
|
reflcl |
|- ( ( B / D ) e. RR -> ( |_ ` ( B / D ) ) e. RR ) |
28 |
27
|
recnd |
|- ( ( B / D ) e. RR -> ( |_ ` ( B / D ) ) e. CC ) |
29 |
26 28
|
syl |
|- ( ( B e. RR /\ D e. RR+ ) -> ( |_ ` ( B / D ) ) e. CC ) |
30 |
25 29
|
mulcld |
|- ( ( B e. RR /\ D e. RR+ ) -> ( D x. ( |_ ` ( B / D ) ) ) e. CC ) |
31 |
30
|
adantrl |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( D x. ( |_ ` ( B / D ) ) ) e. CC ) |
32 |
23 24 31
|
addsubd |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) |
33 |
32
|
adantll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) |
34 |
21 33
|
eqeq12d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) <-> ( ( A - ( D x. ( |_ ` ( A / D ) ) ) ) + C ) = ( ( B - ( D x. ( |_ ` ( B / D ) ) ) ) + C ) ) ) |
35 |
7 34
|
sylibrd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) ) ) |
36 |
|
oveq1 |
|- ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) ) |
37 |
|
readdcl |
|- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR ) |
38 |
37
|
adantrr |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( A + C ) e. RR ) |
39 |
|
simprr |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> D e. RR+ ) |
40 |
14
|
flcld |
|- ( ( A e. RR /\ D e. RR+ ) -> ( |_ ` ( A / D ) ) e. ZZ ) |
41 |
40
|
adantrl |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( |_ ` ( A / D ) ) e. ZZ ) |
42 |
|
modcyc2 |
|- ( ( ( A + C ) e. RR /\ D e. RR+ /\ ( |_ ` ( A / D ) ) e. ZZ ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) ) |
43 |
38 39 41 42
|
syl3anc |
|- ( ( A e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) ) |
44 |
43
|
adantlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( A + C ) mod D ) ) |
45 |
|
readdcl |
|- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR ) |
46 |
45
|
adantrr |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( B + C ) e. RR ) |
47 |
|
simprr |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> D e. RR+ ) |
48 |
26
|
flcld |
|- ( ( B e. RR /\ D e. RR+ ) -> ( |_ ` ( B / D ) ) e. ZZ ) |
49 |
48
|
adantrl |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( |_ ` ( B / D ) ) e. ZZ ) |
50 |
|
modcyc2 |
|- ( ( ( B + C ) e. RR /\ D e. RR+ /\ ( |_ ` ( B / D ) ) e. ZZ ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) ) |
51 |
46 47 49 50
|
syl3anc |
|- ( ( B e. RR /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) ) |
52 |
51
|
adantll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) = ( ( B + C ) mod D ) ) |
53 |
44 52
|
eqeq12d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) mod D ) = ( ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) mod D ) <-> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) ) |
54 |
36 53
|
syl5ib |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( ( A + C ) - ( D x. ( |_ ` ( A / D ) ) ) ) = ( ( B + C ) - ( D x. ( |_ ` ( B / D ) ) ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) ) |
55 |
35 54
|
syld |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) ) -> ( ( A mod D ) = ( B mod D ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) ) |
56 |
55
|
3impia |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR+ ) /\ ( A mod D ) = ( B mod D ) ) -> ( ( A + C ) mod D ) = ( ( B + C ) mod D ) ) |