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Metamath Proof Explorer

Theorem liminfresre

Description: The inferior limit of a function only depends on the real part of its domain. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypothesis liminfresre.1 
|- ( ph -> F e. V )
Assertion liminfresre 
|- ( ph -> ( liminf ` ( F |` RR ) ) = ( liminf ` F ) )

Proof

Step Hyp Ref Expression
1 liminfresre.1 
 |-  ( ph -> F e. V )
2 rge0ssre 
 |-  ( 0 [,) +oo ) C_ RR
3 2 resabs1i 
 |-  ( ( F |` RR ) |` ( 0 [,) +oo ) ) = ( F |` ( 0 [,) +oo ) )
4 3 fveq2i 
 |-  ( liminf ` ( ( F |` RR ) |` ( 0 [,) +oo ) ) ) = ( liminf ` ( F |` ( 0 [,) +oo ) ) )
5 4 a1i 
 |-  ( ph -> ( liminf ` ( ( F |` RR ) |` ( 0 [,) +oo ) ) ) = ( liminf ` ( F |` ( 0 [,) +oo ) ) ) )
6 0red 
 |-  ( ph -> 0 e. RR )
7 eqid 
 |-  ( 0 [,) +oo ) = ( 0 [,) +oo )
8 1 resexd 
 |-  ( ph -> ( F |` RR ) e. _V )
9 6 7 8 liminfresico 
 |-  ( ph -> ( liminf ` ( ( F |` RR ) |` ( 0 [,) +oo ) ) ) = ( liminf ` ( F |` RR ) ) )
10 6 7 1 liminfresico 
 |-  ( ph -> ( liminf ` ( F |` ( 0 [,) +oo ) ) ) = ( liminf ` F ) )
11 5 9 10 3eqtr3d 
 |-  ( ph -> ( liminf ` ( F |` RR ) ) = ( liminf ` F ) )