Step |
Hyp |
Ref |
Expression |
1 |
|
recn |
|- ( B e. RR -> B e. CC ) |
2 |
1
|
3ad2ant2 |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> B e. CC ) |
3 |
|
modcl |
|- ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) e. RR ) |
4 |
3
|
recnd |
|- ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) e. CC ) |
5 |
4
|
3adant2 |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( A mod M ) e. CC ) |
6 |
2 5
|
addcomd |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( B + ( A mod M ) ) = ( ( A mod M ) + B ) ) |
7 |
6
|
oveq1d |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( ( A mod M ) + B ) mod M ) ) |
8 |
|
modaddmod |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( ( A mod M ) + B ) mod M ) = ( ( A + B ) mod M ) ) |
9 |
|
recn |
|- ( A e. RR -> A e. CC ) |
10 |
|
addcom |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) |
11 |
9 1 10
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) ) |
12 |
11
|
oveq1d |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) mod M ) = ( ( B + A ) mod M ) ) |
13 |
12
|
3adant3 |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( A + B ) mod M ) = ( ( B + A ) mod M ) ) |
14 |
7 8 13
|
3eqtrd |
|- ( ( A e. RR /\ B e. RR /\ M e. RR+ ) -> ( ( B + ( A mod M ) ) mod M ) = ( ( B + A ) mod M ) ) |