Metamath Proof Explorer

Theorem liminfval4

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

Ref Expression
Hypotheses liminfval4.x 
|- F/ x ph
liminfval4.a 
|- ( ph -> A e. V )
liminfval4.m 
|- ( ph -> M e. RR )
liminfval4.b 
|- ( ( ph /\ x e. ( A i^i ( M [,) +oo ) ) ) -> B e. RR )
Assertion liminfval4 
|- ( ph -> ( liminf ` ( x e. A |-> B ) ) = -e ( limsup ` ( x e. A |-> -u B ) ) )

Proof

Step Hyp Ref Expression
1 liminfval4.x 
 |-  F/ x ph
2 liminfval4.a 
 |-  ( ph -> A e. V )
3 liminfval4.m 
 |-  ( ph -> M e. RR )
4 liminfval4.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 4 rexrd 
 |-  ( ( ph /\ x e. ( A i^i ( M [,) +oo ) ) ) -> B e. RR* )
9 1 7 8 liminfvalxrmpt 
 |-  ( ph -> ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) )
10 4 rexnegd 
 |-  ( ( ph /\ x e. ( A i^i ( M [,) +oo ) ) ) -> -e B = -u B )
11 1 10 mpteq2da 
 |-  ( ph -> ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) = ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) )
12 11 fveq2d 
 |-  ( ph -> ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) = ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) ) )
13 12 xnegeqd 
 |-  ( ph -> -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -e B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) ) )
14 9 13 eqtrd 
 |-  ( ph -> ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) ) )
15 eqid 
 |-  ( M [,) +oo ) = ( M [,) +oo )
16 3 15 2 liminfresicompt 
 |-  ( ph -> ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) = ( liminf ` ( x e. A |-> B ) ) )
17 16 eqcomd 
 |-  ( ph -> ( liminf ` ( x e. A |-> B ) ) = ( liminf ` ( x e. ( A i^i ( M [,) +oo ) ) |-> B ) ) )
18 2 3 15 limsupresicompt 
 |-  ( ph -> ( limsup ` ( x e. A |-> -u B ) ) = ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) ) )
19 18 xnegeqd 
 |-  ( ph -> -e ( limsup ` ( x e. A |-> -u B ) ) = -e ( limsup ` ( x e. ( A i^i ( M [,) +oo ) ) |-> -u B ) ) )
20 14 17 19 3eqtr4d 
 |-  ( ph -> ( liminf ` ( x e. A |-> B ) ) = -e ( limsup ` ( x e. A |-> -u B ) ) )