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Metamath Proof Explorer

Theorem clim

Description: Express the predicate: The limit of complex number sequence F is A , or F converges to A . This means that for any real x , no matter how small, there always exists an integer j such that the absolute difference of any later complex number in the sequence and the limit is less than x . (Contributed by NM, 28-Aug-2005) (Revised by Mario Carneiro, 28-Apr-2015)

Ref Expression
Hypotheses clim.1 
|- ( ph -> F e. V )
clim.3 
|- ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B )
Assertion clim 
|- ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )

Proof

Step Hyp Ref Expression
1 clim.1 
 |-  ( ph -> F e. V )
2 clim.3 
 |-  ( ( ph /\ k e. ZZ ) -> ( F ` k ) = B )
3 climrel 
 |-  Rel ~~>
4 3 brrelex2i 
 |-  ( F ~~> A -> A e. _V )
5 4 a1i 
 |-  ( ph -> ( F ~~> A -> A e. _V ) )
6 elex 
 |-  ( A e. CC -> A e. _V )
7 6 adantr 
 |-  ( ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) -> A e. _V )
8 7 a1i 
 |-  ( ph -> ( ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) -> A e. _V ) )
9 simpr 
 |-  ( ( f = F /\ y = A ) -> y = A )
10 9 eleq1d 
 |-  ( ( f = F /\ y = A ) -> ( y e. CC <-> A e. CC ) )
11 fveq1 
 |-  ( f = F -> ( f ` k ) = ( F ` k ) )
12 11 adantr 
 |-  ( ( f = F /\ y = A ) -> ( f ` k ) = ( F ` k ) )
13 12 eleq1d 
 |-  ( ( f = F /\ y = A ) -> ( ( f ` k ) e. CC <-> ( F ` k ) e. CC ) )
14 oveq12 
 |-  ( ( ( f ` k ) = ( F ` k ) /\ y = A ) -> ( ( f ` k ) - y ) = ( ( F ` k ) - A ) )
15 11 14 sylan 
 |-  ( ( f = F /\ y = A ) -> ( ( f ` k ) - y ) = ( ( F ` k ) - A ) )
16 15 fveq2d 
 |-  ( ( f = F /\ y = A ) -> ( abs ` ( ( f ` k ) - y ) ) = ( abs ` ( ( F ` k ) - A ) ) )
17 16 breq1d 
 |-  ( ( f = F /\ y = A ) -> ( ( abs ` ( ( f ` k ) - y ) ) < x <-> ( abs ` ( ( F ` k ) - A ) ) < x ) )
18 13 17 anbi12d 
 |-  ( ( f = F /\ y = A ) -> ( ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) <-> ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
19 18 ralbidv 
 |-  ( ( f = F /\ y = A ) -> ( A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) <-> A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
20 19 rexbidv 
 |-  ( ( f = F /\ y = A ) -> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
21 20 ralbidv 
 |-  ( ( f = F /\ y = A ) -> ( A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) <-> A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) )
22 10 21 anbi12d 
 |-  ( ( f = F /\ y = A ) -> ( ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
23 df-clim 
 |-  ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
24 22 23 brabga 
 |-  ( ( F e. V /\ A e. _V ) -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
25 24 ex 
 |-  ( F e. V -> ( A e. _V -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) ) )
26 1 25 syl 
 |-  ( ph -> ( A e. _V -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) ) )
27 5 8 26 pm5.21ndd 
 |-  ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) ) )
28 eluzelz 
 |-  ( k e. ( ZZ>= ` j ) -> k e. ZZ )
29 2 eleq1d 
 |-  ( ( ph /\ k e. ZZ ) -> ( ( F ` k ) e. CC <-> B e. CC ) )
30 2 fvoveq1d 
 |-  ( ( ph /\ k e. ZZ ) -> ( abs ` ( ( F ` k ) - A ) ) = ( abs ` ( B - A ) ) )
31 30 breq1d 
 |-  ( ( ph /\ k e. ZZ ) -> ( ( abs ` ( ( F ` k ) - A ) ) < x <-> ( abs ` ( B - A ) ) < x ) )
32 29 31 anbi12d 
 |-  ( ( ph /\ k e. ZZ ) -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
33 28 32 sylan2 
 |-  ( ( ph /\ k e. ( ZZ>= ` j ) ) -> ( ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
34 33 ralbidva 
 |-  ( ph -> ( A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
35 34 rexbidv 
 |-  ( ph -> ( E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
36 35 ralbidv 
 |-  ( ph -> ( A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) <-> A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) )
37 36 anbi2d 
 |-  ( ph -> ( ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( F ` k ) e. CC /\ ( abs ` ( ( F ` k ) - A ) ) < x ) ) <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )
38 27 37 bitrd 
 |-  ( ph -> ( F ~~> A <-> ( A e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( B e. CC /\ ( abs ` ( B - A ) ) < x ) ) ) )