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Metamath Proof Explorer

Theorem limsupvaluz3

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 limsupvaluz3.k 
|- F/ k ph
limsupvaluz3.m 
|- ( ph -> M e. ZZ )
limsupvaluz3.z 
|- Z = ( ZZ>= ` M )
limsupvaluz3.b 
|- ( ( ph /\ k e. Z ) -> B e. RR* )
Assertion limsupvaluz3 
|- ( ph -> ( limsup ` ( k e. Z |-> B ) ) = -e ( liminf ` ( k e. Z |-> -e B ) ) )

Proof

Step Hyp Ref Expression
1 limsupvaluz3.k 
 |-  F/ k ph
2 limsupvaluz3.m 
 |-  ( ph -> M e. ZZ )
3 limsupvaluz3.z 
 |-  Z = ( ZZ>= ` M )
4 limsupvaluz3.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 limsupval4 
 |-  ( ph -> ( limsup ` ( k e. Z |-> B ) ) = -e ( liminf ` ( k e. Z |-> -e B ) ) )