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Metamath Proof Explorer

Theorem liminfgval

Description: Value of the inferior limit function. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypothesis liminfgval.1 
|- G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
Assertion liminfgval 
|- ( M e. RR -> ( G ` M ) = inf ( ( ( F " ( M [,) +oo ) ) i^i RR* ) , RR* , < ) )

Proof

Step Hyp Ref Expression
1 liminfgval.1 
 |-  G = ( k e. RR |-> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
2 oveq1 
 |-  ( k = M -> ( k [,) +oo ) = ( M [,) +oo ) )
3 2 imaeq2d 
 |-  ( k = M -> ( F " ( k [,) +oo ) ) = ( F " ( M [,) +oo ) ) )
4 3 ineq1d 
 |-  ( k = M -> ( ( F " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( M [,) +oo ) ) i^i RR* ) )
5 4 infeq1d 
 |-  ( k = M -> inf ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = inf ( ( ( F " ( M [,) +oo ) ) i^i RR* ) , RR* , < ) )
6 xrltso 
 |-  < Or RR*
7 6 infex 
 |-  inf ( ( ( F " ( M [,) +oo ) ) i^i RR* ) , RR* , < ) e. _V
8 5 1 7 fvmpt 
 |-  ( M e. RR -> ( G ` M ) = inf ( ( ( F " ( M [,) +oo ) ) i^i RR* ) , RR* , < ) )