Step |
Hyp |
Ref |
Expression |
1 |
|
fladdz |
|- ( ( A e. RR /\ N e. ZZ ) -> ( |_ ` ( A + N ) ) = ( ( |_ ` A ) + N ) ) |
2 |
|
recn |
|- ( A e. RR -> A e. CC ) |
3 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
4 |
|
addcom |
|- ( ( A e. CC /\ N e. CC ) -> ( A + N ) = ( N + A ) ) |
5 |
2 3 4
|
syl2an |
|- ( ( A e. RR /\ N e. ZZ ) -> ( A + N ) = ( N + A ) ) |
6 |
5
|
fveq2d |
|- ( ( A e. RR /\ N e. ZZ ) -> ( |_ ` ( A + N ) ) = ( |_ ` ( N + A ) ) ) |
7 |
|
reflcl |
|- ( A e. RR -> ( |_ ` A ) e. RR ) |
8 |
7
|
recnd |
|- ( A e. RR -> ( |_ ` A ) e. CC ) |
9 |
|
addcom |
|- ( ( ( |_ ` A ) e. CC /\ N e. CC ) -> ( ( |_ ` A ) + N ) = ( N + ( |_ ` A ) ) ) |
10 |
8 3 9
|
syl2an |
|- ( ( A e. RR /\ N e. ZZ ) -> ( ( |_ ` A ) + N ) = ( N + ( |_ ` A ) ) ) |
11 |
1 6 10
|
3eqtr3d |
|- ( ( A e. RR /\ N e. ZZ ) -> ( |_ ` ( N + A ) ) = ( N + ( |_ ` A ) ) ) |
12 |
11
|
ancoms |
|- ( ( N e. ZZ /\ A e. RR ) -> ( |_ ` ( N + A ) ) = ( N + ( |_ ` A ) ) ) |