Step |
Hyp |
Ref |
Expression |
1 |
|
climabs0.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
climabs0.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
climabs0.3 |
|- ( ph -> F e. V ) |
4 |
|
climabs0.4 |
|- ( ph -> G e. W ) |
5 |
|
climabs0.5 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC ) |
6 |
|
climabs0.6 |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) = ( abs ` ( F ` k ) ) ) |
7 |
1
|
uztrn2 |
|- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z ) |
8 |
|
absidm |
|- ( ( F ` k ) e. CC -> ( abs ` ( abs ` ( F ` k ) ) ) = ( abs ` ( F ` k ) ) ) |
9 |
5 8
|
syl |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( abs ` ( F ` k ) ) ) = ( abs ` ( F ` k ) ) ) |
10 |
9
|
breq1d |
|- ( ( ph /\ k e. Z ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) ) |
11 |
7 10
|
sylan2 |
|- ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) ) |
12 |
11
|
anassrs |
|- ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( abs ` ( F ` k ) ) ) < x <-> ( abs ` ( F ` k ) ) < x ) ) |
13 |
12
|
ralbidva |
|- ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( abs ` ( F ` k ) ) ) < x <-> A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < x ) ) |
14 |
13
|
rexbidva |
|- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( abs ` ( F ` k ) ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < x ) ) |
15 |
14
|
ralbidv |
|- ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( abs ` ( F ` k ) ) ) < x <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < x ) ) |
16 |
5
|
abscld |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. RR ) |
17 |
16
|
recnd |
|- ( ( ph /\ k e. Z ) -> ( abs ` ( F ` k ) ) e. CC ) |
18 |
1 2 4 6 17
|
clim0c |
|- ( ph -> ( G ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( abs ` ( F ` k ) ) ) < x ) ) |
19 |
|
eqidd |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( F ` k ) ) |
20 |
1 2 3 19 5
|
clim0c |
|- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( F ` k ) ) < x ) ) |
21 |
15 18 20
|
3bitr4rd |
|- ( ph -> ( F ~~> 0 <-> G ~~> 0 ) ) |