Metamath Proof Explorer


Theorem liminfvaluz

Description: Alternate definition of liminf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses liminfvaluz.k
|- F/ k ph
liminfvaluz.m
|- ( ph -> M e. ZZ )
liminfvaluz.z
|- Z = ( ZZ>= ` M )
liminfvaluz.b
|- ( ( ph /\ k e. Z ) -> B e. RR* )
Assertion liminfvaluz
|- ( ph -> ( liminf ` ( k e. Z |-> B ) ) = -e ( limsup ` ( k e. Z |-> -e B ) ) )

Proof

Step Hyp Ref Expression
1 liminfvaluz.k
 |-  F/ k ph
2 liminfvaluz.m
 |-  ( ph -> M e. ZZ )
3 liminfvaluz.z
 |-  Z = ( ZZ>= ` M )
4 liminfvaluz.b
 |-  ( ( ph /\ k e. Z ) -> B e. RR* )
5 3 fvexi
 |-  Z e. _V
6 5 a1i
 |-  ( ph -> Z e. _V )
7 2 zred
 |-  ( ph -> M e. RR )
8 simpr
 |-  ( ( ph /\ k e. ( Z i^i ( M [,) +oo ) ) ) -> k e. ( Z i^i ( M [,) +oo ) ) )
9 2 3 uzinico3
 |-  ( ph -> Z = ( Z i^i ( M [,) +oo ) ) )
10 9 eqcomd
 |-  ( ph -> ( Z i^i ( M [,) +oo ) ) = Z )
11 10 adantr
 |-  ( ( ph /\ k e. ( Z i^i ( M [,) +oo ) ) ) -> ( Z i^i ( M [,) +oo ) ) = Z )
12 8 11 eleqtrd
 |-  ( ( ph /\ k e. ( Z i^i ( M [,) +oo ) ) ) -> k e. Z )
13 12 4 syldan
 |-  ( ( ph /\ k e. ( Z i^i ( M [,) +oo ) ) ) -> B e. RR* )
14 1 6 7 13 liminfval3
 |-  ( ph -> ( liminf ` ( k e. Z |-> B ) ) = -e ( limsup ` ( k e. Z |-> -e B ) ) )