Step |
Hyp |
Ref |
Expression |
1 |
|
hgt750lemc.n |
|- ( ph -> N e. NN ) |
2 |
1
|
nnzd |
|- ( ph -> N e. ZZ ) |
3 |
|
chpvalz |
|- ( N e. ZZ -> ( psi ` N ) = sum_ j e. ( 1 ... N ) ( Lam ` j ) ) |
4 |
2 3
|
syl |
|- ( ph -> ( psi ` N ) = sum_ j e. ( 1 ... N ) ( Lam ` j ) ) |
5 |
|
fveq2 |
|- ( x = N -> ( psi ` x ) = ( psi ` N ) ) |
6 |
|
oveq2 |
|- ( x = N -> ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. x ) = ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. N ) ) |
7 |
5 6
|
breq12d |
|- ( x = N -> ( ( psi ` x ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. x ) <-> ( psi ` N ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. N ) ) ) |
8 |
|
ax-ros335 |
|- A. x e. RR+ ( psi ` x ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. x ) |
9 |
8
|
a1i |
|- ( ph -> A. x e. RR+ ( psi ` x ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. x ) ) |
10 |
1
|
nnrpd |
|- ( ph -> N e. RR+ ) |
11 |
7 9 10
|
rspcdva |
|- ( ph -> ( psi ` N ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. N ) ) |
12 |
4 11
|
eqbrtrrd |
|- ( ph -> sum_ j e. ( 1 ... N ) ( Lam ` j ) < ( ( 1 . _ 0 _ 3 _ 8 _ 8 3 ) x. N ) ) |