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Metamath Proof Explorer

Theorem clim0cf

Description: Express the predicate F converges to 0 . Similar to clim , but without the disjoint var constraint F k . (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses clim0cf.nf 
|- F/_ k F
clim0cf.z 
|- Z = ( ZZ>= ` M )
clim0cf.m 
|- ( ph -> M e. ZZ )
clim0cf.f 
|- ( ph -> F e. V )
clim0cf.fv 
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
clim0cf.b 
|- ( ( ph /\ k e. Z ) -> B e. CC )
Assertion clim0cf 
|- ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )

Proof

Step Hyp Ref Expression
1 clim0cf.nf 
 |-  F/_ k F
2 clim0cf.z 
 |-  Z = ( ZZ>= ` M )
3 clim0cf.m 
 |-  ( ph -> M e. ZZ )
4 clim0cf.f 
 |-  ( ph -> F e. V )
5 clim0cf.fv 
 |-  ( ( ph /\ k e. Z ) -> ( F ` k ) = B )
6 clim0cf.b 
 |-  ( ( ph /\ k e. Z ) -> B e. CC )
7 0cnd 
 |-  ( ph -> 0 e. CC )
8 1 2 3 4 5 7 6 clim2cf 
 |-  ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - 0 ) ) < x ) )
9 2 uztrn2 
 |-  ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
10 6 subid1d 
 |-  ( ( ph /\ k e. Z ) -> ( B - 0 ) = B )
11 10 fveq2d 
 |-  ( ( ph /\ k e. Z ) -> ( abs ` ( B - 0 ) ) = ( abs ` B ) )
12 11 breq1d 
 |-  ( ( ph /\ k e. Z ) -> ( ( abs ` ( B - 0 ) ) < x <-> ( abs ` B ) < x ) )
13 9 12 sylan2 
 |-  ( ( ph /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( B - 0 ) ) < x <-> ( abs ` B ) < x ) )
14 13 anassrs 
 |-  ( ( ( ph /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( B - 0 ) ) < x <-> ( abs ` B ) < x ) )
15 14 ralbidva 
 |-  ( ( ph /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( B - 0 ) ) < x <-> A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
16 15 rexbidva 
 |-  ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - 0 ) ) < x <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
17 16 ralbidv 
 |-  ( ph -> ( A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( B - 0 ) ) < x <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )
18 8 17 bitrd 
 |-  ( ph -> ( F ~~> 0 <-> A. x e. RR+ E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` B ) < x ) )