This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem liminfval3

Description: Alternate definition of liminf when the given function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses liminfval3.x 
|- F/ x ph
liminfval3.a 
|- ( ph -> A e. V )
liminfval3.m 
|- ( ph -> M e. RR )
liminfval3.b 
|- ( ( ph /\ x e. ( A i^i ( M [,) +oo ) ) ) -> B e. RR* )
Assertion liminfval3 
|- ( ph -> ( liminf ` ( x e. A |-> B ) ) = -e ( limsup ` ( x e. A |-> -e B ) ) )

Proof

Step Hyp Ref Expression
1 liminfval3.x 
 |-  F/ x ph
2 liminfval3.a 
 |-  ( ph -> A e. V )
3 liminfval3.m 
 |-  ( ph -> M e. RR )
4 liminfval3.b 
 |-  ( ( ph /\ x e. ( A i^i ( M [,) +oo ) ) ) -> B e. RR* )
5 inss1 
 |-  ( A i^i ( M [,) +oo ) ) C_ A
6 5 a1i 
 |-  ( ph -> ( A i^i ( M [,) +oo ) ) C_ A )
7 2 6 ssexd 
 |-  ( ph -> ( A i^i ( M [,) +oo ) ) e. _V )
8 1 7 4 liminfvalxrmpt 
 |-  ( ph -> ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) )
9 eqid 
 |-  ( M [,) +oo ) = ( M [,) +oo )
10 3 9 2 liminfresicompt 
 |-  ( ph -> ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) = ( liminf ` ( x e. A |-> B ) ) )
11 10 eqcomd 
 |-  ( ph -> ( liminf ` ( x e. A |-> B ) ) = ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) )
12 2 3 9 limsupresicompt 
 |-  ( ph -> ( limsup ` ( x e. A |-> -e B ) ) = ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) )
13 12 xnegeqd 
 |-  ( ph -> -e ( limsup ` ( x e. A |-> -e B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) )
14 8 11 13 3eqtr4d 
 |-  ( ph -> ( liminf ` ( x e. A |-> B ) ) = -e ( limsup ` ( x e. A |-> -e B ) ) )