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Metamath Proof Explorer

Theorem chlimi

Description: The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999) (New usage is discouraged.)

Ref Expression
Hypothesis chlim.1 
|- A e. _V
Assertion chlimi 
|- ( ( H e. CH /\ F : NN --> H /\ F ~~>v A ) -> A e. H )

Proof

Step Hyp Ref Expression
1 chlim.1 
 |-  A e. _V
2 isch2 
 |-  ( H e. CH <-> ( H e. SH /\ A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) ) )
3 2 simprbi 
 |-  ( H e. CH -> A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) )
4 nnex 
 |-  NN e. _V
5 fex 
 |-  ( ( F : NN --> H /\ NN e. _V ) -> F e. _V )
6 4 5 mpan2 
 |-  ( F : NN --> H -> F e. _V )
7 6 adantr 
 |-  ( ( F : NN --> H /\ F ~~>v A ) -> F e. _V )
8 feq1 
 |-  ( f = F -> ( f : NN --> H <-> F : NN --> H ) )
9 breq1 
 |-  ( f = F -> ( f ~~>v x <-> F ~~>v x ) )
10 8 9 anbi12d 
 |-  ( f = F -> ( ( f : NN --> H /\ f ~~>v x ) <-> ( F : NN --> H /\ F ~~>v x ) ) )
11 10 imbi1d 
 |-  ( f = F -> ( ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) <-> ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
12 11 albidv 
 |-  ( f = F -> ( A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) <-> A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
13 12 spcgv 
 |-  ( F e. _V -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) ) )
14 breq2 
 |-  ( x = A -> ( F ~~>v x <-> F ~~>v A ) )
15 14 anbi2d 
 |-  ( x = A -> ( ( F : NN --> H /\ F ~~>v x ) <-> ( F : NN --> H /\ F ~~>v A ) ) )
16 eleq1 
 |-  ( x = A -> ( x e. H <-> A e. H ) )
17 15 16 imbi12d 
 |-  ( x = A -> ( ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) <-> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
18 1 17 spcv 
 |-  ( A. x ( ( F : NN --> H /\ F ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
19 13 18 syl6 
 |-  ( F e. _V -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
20 7 19 syl 
 |-  ( ( F : NN --> H /\ F ~~>v A ) -> ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) ) )
21 20 pm2.43b 
 |-  ( A. f A. x ( ( f : NN --> H /\ f ~~>v x ) -> x e. H ) -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
22 3 21 syl 
 |-  ( H e. CH -> ( ( F : NN --> H /\ F ~~>v A ) -> A e. H ) )
23 22 3impib 
 |-  ( ( H e. CH /\ F : NN --> H /\ F ~~>v A ) -> A e. H )