Step |
Hyp |
Ref |
Expression |
1 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
2 |
|
rpre |
|- ( B e. RR+ -> B e. RR ) |
3 |
|
remulcl |
|- ( ( N e. RR /\ B e. RR ) -> ( N x. B ) e. RR ) |
4 |
1 2 3
|
syl2an |
|- ( ( N e. ZZ /\ B e. RR+ ) -> ( N x. B ) e. RR ) |
5 |
|
readdcl |
|- ( ( A e. RR /\ ( N x. B ) e. RR ) -> ( A + ( N x. B ) ) e. RR ) |
6 |
4 5
|
sylan2 |
|- ( ( A e. RR /\ ( N e. ZZ /\ B e. RR+ ) ) -> ( A + ( N x. B ) ) e. RR ) |
7 |
6
|
3impb |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( A + ( N x. B ) ) e. RR ) |
8 |
|
simp3 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> B e. RR+ ) |
9 |
|
modval |
|- ( ( ( A + ( N x. B ) ) e. RR /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) mod B ) = ( ( A + ( N x. B ) ) - ( B x. ( |_ ` ( ( A + ( N x. B ) ) / B ) ) ) ) ) |
10 |
7 8 9
|
syl2anc |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) mod B ) = ( ( A + ( N x. B ) ) - ( B x. ( |_ ` ( ( A + ( N x. B ) ) / B ) ) ) ) ) |
11 |
|
recn |
|- ( A e. RR -> A e. CC ) |
12 |
11
|
3ad2ant1 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> A e. CC ) |
13 |
4
|
recnd |
|- ( ( N e. ZZ /\ B e. RR+ ) -> ( N x. B ) e. CC ) |
14 |
13
|
3adant1 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( N x. B ) e. CC ) |
15 |
|
rpcnne0 |
|- ( B e. RR+ -> ( B e. CC /\ B =/= 0 ) ) |
16 |
15
|
3ad2ant3 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B e. CC /\ B =/= 0 ) ) |
17 |
|
divdir |
|- ( ( A e. CC /\ ( N x. B ) e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A + ( N x. B ) ) / B ) = ( ( A / B ) + ( ( N x. B ) / B ) ) ) |
18 |
12 14 16 17
|
syl3anc |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) / B ) = ( ( A / B ) + ( ( N x. B ) / B ) ) ) |
19 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
20 |
|
divcan4 |
|- ( ( N e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( N x. B ) / B ) = N ) |
21 |
20
|
3expb |
|- ( ( N e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( N x. B ) / B ) = N ) |
22 |
19 15 21
|
syl2an |
|- ( ( N e. ZZ /\ B e. RR+ ) -> ( ( N x. B ) / B ) = N ) |
23 |
22
|
3adant1 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( N x. B ) / B ) = N ) |
24 |
23
|
oveq2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A / B ) + ( ( N x. B ) / B ) ) = ( ( A / B ) + N ) ) |
25 |
18 24
|
eqtrd |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) / B ) = ( ( A / B ) + N ) ) |
26 |
25
|
fveq2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( |_ ` ( ( A + ( N x. B ) ) / B ) ) = ( |_ ` ( ( A / B ) + N ) ) ) |
27 |
|
rerpdivcl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR ) |
28 |
27
|
3adant2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( A / B ) e. RR ) |
29 |
|
simp2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> N e. ZZ ) |
30 |
|
fladdz |
|- ( ( ( A / B ) e. RR /\ N e. ZZ ) -> ( |_ ` ( ( A / B ) + N ) ) = ( ( |_ ` ( A / B ) ) + N ) ) |
31 |
28 29 30
|
syl2anc |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( |_ ` ( ( A / B ) + N ) ) = ( ( |_ ` ( A / B ) ) + N ) ) |
32 |
26 31
|
eqtrd |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( |_ ` ( ( A + ( N x. B ) ) / B ) ) = ( ( |_ ` ( A / B ) ) + N ) ) |
33 |
32
|
oveq2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B x. ( |_ ` ( ( A + ( N x. B ) ) / B ) ) ) = ( B x. ( ( |_ ` ( A / B ) ) + N ) ) ) |
34 |
|
rpcn |
|- ( B e. RR+ -> B e. CC ) |
35 |
34
|
3ad2ant3 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> B e. CC ) |
36 |
|
reflcl |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. RR ) |
37 |
36
|
recnd |
|- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) e. CC ) |
38 |
27 37
|
syl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. CC ) |
39 |
38
|
3adant2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( |_ ` ( A / B ) ) e. CC ) |
40 |
19
|
3ad2ant2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> N e. CC ) |
41 |
35 39 40
|
adddid |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B x. ( ( |_ ` ( A / B ) ) + N ) ) = ( ( B x. ( |_ ` ( A / B ) ) ) + ( B x. N ) ) ) |
42 |
|
mulcom |
|- ( ( N e. CC /\ B e. CC ) -> ( N x. B ) = ( B x. N ) ) |
43 |
19 34 42
|
syl2an |
|- ( ( N e. ZZ /\ B e. RR+ ) -> ( N x. B ) = ( B x. N ) ) |
44 |
43
|
3adant1 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( N x. B ) = ( B x. N ) ) |
45 |
44
|
eqcomd |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B x. N ) = ( N x. B ) ) |
46 |
45
|
oveq2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( B x. ( |_ ` ( A / B ) ) ) + ( B x. N ) ) = ( ( B x. ( |_ ` ( A / B ) ) ) + ( N x. B ) ) ) |
47 |
33 41 46
|
3eqtrd |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B x. ( |_ ` ( ( A + ( N x. B ) ) / B ) ) ) = ( ( B x. ( |_ ` ( A / B ) ) ) + ( N x. B ) ) ) |
48 |
47
|
oveq2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) - ( B x. ( |_ ` ( ( A + ( N x. B ) ) / B ) ) ) ) = ( ( A + ( N x. B ) ) - ( ( B x. ( |_ ` ( A / B ) ) ) + ( N x. B ) ) ) ) |
49 |
34
|
adantl |
|- ( ( A e. RR /\ B e. RR+ ) -> B e. CC ) |
50 |
49 38
|
mulcld |
|- ( ( A e. RR /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC ) |
51 |
50
|
3adant2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( B x. ( |_ ` ( A / B ) ) ) e. CC ) |
52 |
12 51 14
|
pnpcan2d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) - ( ( B x. ( |_ ` ( A / B ) ) ) + ( N x. B ) ) ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) |
53 |
10 48 52
|
3eqtrd |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) |
54 |
|
modval |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) |
55 |
54
|
3adant2 |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( A mod B ) = ( A - ( B x. ( |_ ` ( A / B ) ) ) ) ) |
56 |
53 55
|
eqtr4d |
|- ( ( A e. RR /\ N e. ZZ /\ B e. RR+ ) -> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) ) |
57 |
56
|
3com23 |
|- ( ( A e. RR /\ B e. RR+ /\ N e. ZZ ) -> ( ( A + ( N x. B ) ) mod B ) = ( A mod B ) ) |