Metamath Proof Explorer

Theorem clim2c

Description: Express the predicate F converges to A . (Contributed by NM, 24-Feb-2008) (Revised by Mario Carneiro, 31-Jan-2014)

Ref Expression
Hypotheses clim2.1 
|- Z = ( ZZ>= ` M )
clim2.2 
|- ( ph -> M e. ZZ )
clim2.3 
|- ( ph -> F e. V )
clim2.4 
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
clim2c.5 
|- ( ph -> A e. CC )
clim2c.6 
|- ( ( ph /\ k e. Z ) -> B e. CC )
Assertion clim2c 
|- ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )

Proof

Step Hyp Ref Expression
1 clim2.1 
 |-  Z = ( ZZ>= ` M )
2 clim2.2 
 |-  ( ph -> M e. ZZ )
3 clim2.3 
 |-  ( ph -> F e. V )
4 clim2.4 
 |-  ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
5 clim2c.5 
 |-  ( ph -> A e. CC )
6 clim2c.6 
 |-  ( ( ph /\ k e. Z ) -> B e. CC )
7 5 biantrurd 
 |-  ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
8 1 uztrn2 
 |-  ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
9 6 biantrurd 
 |-  ( ( ph /\ k e. Z ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
10 8 9 sylan2 
 |-  ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
11 10 anassrs 
 |-  ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( B - A ) ) < x <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
12 11 ralbidva 
 |-  ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x <-> A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
13 12 rexbidva 
 |-  ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
14 13 ralbidv 
 |-  ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
15 1 2 3 4 clim2 
 |-  ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
16 7 14 15 3bitr4rd 
 |-  ( ph -> ( F ~~> A <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - A ) ) < x ) )