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Metamath Proof Explorer

Theorem limsupresicompt

Description: The superior limit doesn't change when a function is restricted to the upper part of the reals. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses limsupresicompt.a 
|- ( ph -> A e. V )
limsupresicompt.m 
|- ( ph -> M e. RR )
limsupresicompt.z 
|- Z = ( M [,) +oo )
Assertion limsupresicompt 
|- ( ph -> ( limsup ` ( x e. A |-> B ) ) = ( limsup ` ( x e. ( A i^i Z ) |-> B ) ) )

Proof

Step Hyp Ref Expression
1 limsupresicompt.a 
 |-  ( ph -> A e. V )
2 limsupresicompt.m 
 |-  ( ph -> M e. RR )
3 limsupresicompt.z 
 |-  Z = ( M [,) +oo )
4 1 mptexd 
 |-  ( ph -> ( x e. A |-> B ) e. _V )
5 2 3 4 limsupresico 
 |-  ( ph -> ( limsup ` ( ( x e. A |-> B ) |` Z ) ) = ( limsup ` ( x e. A |-> B ) ) )
6 resmpt3 
 |-  ( ( x e. A |-> B ) |` Z ) = ( x e. ( A i^i Z ) |-> B )
7 6 a1i 
 |-  ( ph -> ( ( x e. A |-> B ) |` Z ) = ( x e. ( A i^i Z ) |-> B ) )
8 7 fveq2d 
 |-  ( ph -> ( limsup ` ( ( x e. A |-> B ) |` Z ) ) = ( limsup ` ( x e. ( A i^i Z ) |-> B ) ) )
9 5 8 eqtr3d 
 |-  ( ph -> ( limsup ` ( x e. A |-> B ) ) = ( limsup ` ( x e. ( A i^i Z ) |-> B ) ) )