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Metamath Proof Explorer

Theorem hhcmpl

Description: Lemma used for derivation of the completeness axiom ax-hcompl from ZFC Hilbert space theory. (Contributed by NM, 9-Apr-2008) (New usage is discouraged.)

Ref Expression
Hypotheses hhlm.1 
|- U = <. <. +h , .h >. , normh >.
hhlm.2 
|- D = ( IndMet ` U )
hhlm.3 
|- J = ( MetOpen ` D )
hhcmpl.c 
|- ( F e. ( Cau ` D ) -> E. x e. ~H F ( ~~>t ` J ) x )
Assertion hhcmpl 
|- ( F e. Cauchy -> E. x e. ~H F ~~>v x )

Proof

Step Hyp Ref Expression
1 hhlm.1 
 |-  U = <. <. +h , .h >. , normh >.
2 hhlm.2 
 |-  D = ( IndMet ` U )
3 hhlm.3 
 |-  J = ( MetOpen ` D )
4 hhcmpl.c 
 |-  ( F e. ( Cau ` D ) -> E. x e. ~H F ( ~~>t ` J ) x )
5 4 anim1ci 
 |-  ( ( F e. ( Cau ` D ) /\ F e. ( ~H ^m NN ) ) -> ( F e. ( ~H ^m NN ) /\ E. x e. ~H F ( ~~>t ` J ) x ) )
6 elin 
 |-  ( F e. ( ( Cau ` D ) i^i ( ~H ^m NN ) ) <-> ( F e. ( Cau ` D ) /\ F e. ( ~H ^m NN ) ) )
7 r19.42v 
 |-  ( E. x e. ~H ( F e. ( ~H ^m NN ) /\ F ( ~~>t ` J ) x ) <-> ( F e. ( ~H ^m NN ) /\ E. x e. ~H F ( ~~>t ` J ) x ) )
8 5 6 7 3imtr4i 
 |-  ( F e. ( ( Cau ` D ) i^i ( ~H ^m NN ) ) -> E. x e. ~H ( F e. ( ~H ^m NN ) /\ F ( ~~>t ` J ) x ) )
9 1 2 hhcau 
 |-  Cauchy = ( ( Cau ` D ) i^i ( ~H ^m NN ) )
10 9 eleq2i 
 |-  ( F e. Cauchy <-> F e. ( ( Cau ` D ) i^i ( ~H ^m NN ) ) )
11 1 2 3 hhlm 
 |-  ~~>v = ( ( ~~>t ` J ) |` ( ~H ^m NN ) )
12 11 breqi 
 |-  ( F ~~>v x <-> F ( ( ~~>t ` J ) |` ( ~H ^m NN ) ) x )
13 vex 
 |-  x e. _V
14 13 brresi 
 |-  ( F ( ( ~~>t ` J ) |` ( ~H ^m NN ) ) x <-> ( F e. ( ~H ^m NN ) /\ F ( ~~>t ` J ) x ) )
15 12 14 bitri 
 |-  ( F ~~>v x <-> ( F e. ( ~H ^m NN ) /\ F ( ~~>t ` J ) x ) )
16 15 rexbii 
 |-  ( E. x e. ~H F ~~>v x <-> E. x e. ~H ( F e. ( ~H ^m NN ) /\ F ( ~~>t ` J ) x ) )
17 8 10 16 3imtr4i 
 |-  ( F e. Cauchy -> E. x e. ~H F ~~>v x )