Step |
Hyp |
Ref |
Expression |
1 |
|
zre |
|- ( A e. ZZ -> A e. RR ) |
2 |
1
|
adantr |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> A e. RR ) |
3 |
|
ppival |
|- ( A e. RR -> ( ppi ` A ) = ( # ` ( ( 0 [,] A ) i^i Prime ) ) ) |
4 |
2 3
|
syl |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( ppi ` A ) = ( # ` ( ( 0 [,] A ) i^i Prime ) ) ) |
5 |
|
ppisval2 |
|- ( ( A e. RR /\ 2 e. ( ZZ>= ` M ) ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( M ... ( |_ ` A ) ) i^i Prime ) ) |
6 |
1 5
|
sylan |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( M ... ( |_ ` A ) ) i^i Prime ) ) |
7 |
|
flid |
|- ( A e. ZZ -> ( |_ ` A ) = A ) |
8 |
7
|
oveq2d |
|- ( A e. ZZ -> ( M ... ( |_ ` A ) ) = ( M ... A ) ) |
9 |
8
|
ineq1d |
|- ( A e. ZZ -> ( ( M ... ( |_ ` A ) ) i^i Prime ) = ( ( M ... A ) i^i Prime ) ) |
10 |
9
|
adantr |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( ( M ... ( |_ ` A ) ) i^i Prime ) = ( ( M ... A ) i^i Prime ) ) |
11 |
6 10
|
eqtrd |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( M ... A ) i^i Prime ) ) |
12 |
11
|
fveq2d |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( # ` ( ( 0 [,] A ) i^i Prime ) ) = ( # ` ( ( M ... A ) i^i Prime ) ) ) |
13 |
4 12
|
eqtrd |
|- ( ( A e. ZZ /\ 2 e. ( ZZ>= ` M ) ) -> ( ppi ` A ) = ( # ` ( ( M ... A ) i^i Prime ) ) ) |