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Theorem chlimi 25975
Description: The limit property of a closed subspace of a Hilbert space. (Contributed by NM, 14-Sep-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chlim.1  |-  A  e. 
_V
Assertion
Ref Expression
chlimi  |-  ( ( H  e.  CH  /\  F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H )

Proof of Theorem chlimi
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isch2 25964 . . . 4  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
21simprbi 464 . . 3  |-  ( H  e.  CH  ->  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
3 nnex 10554 . . . . . . 7  |-  NN  e.  _V
4 fex 6144 . . . . . . 7  |-  ( ( F : NN --> H  /\  NN  e.  _V )  ->  F  e.  _V )
53, 4mpan2 671 . . . . . 6  |-  ( F : NN --> H  ->  F  e.  _V )
65adantr 465 . . . . 5  |-  ( ( F : NN --> H  /\  F  ~~>v  A )  ->  F  e.  _V )
7 feq1 5719 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f : NN --> H  <->  F : NN
--> H ) )
8 breq1 4456 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  ~~>v  x  <->  F  ~~>v  x ) )
97, 8anbi12d 710 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f : NN --> H  /\  f  ~~>v  x )  <-> 
( F : NN --> H  /\  F  ~~>v  x ) ) )
109imbi1d 317 . . . . . . . 8  |-  ( f  =  F  ->  (
( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( ( F : NN --> H  /\  F  ~~>v  x )  ->  x  e.  H )
) )
1110albidv 1689 . . . . . . 7  |-  ( f  =  F  ->  ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  A. x ( ( F : NN --> H  /\  F  ~~>v  x )  ->  x  e.  H )
) )
1211spcgv 3203 . . . . . 6  |-  ( 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 ) ) )
13 chlim.1 . . . . . . 7  |-  A  e. 
_V
14 breq2 4457 . . . . . . . . 9  |-  ( x  =  A  ->  ( F  ~~>v  x  <->  F  ~~>v  A ) )
1514anbi2d 703 . . . . . . . 8  |-  ( x  =  A  ->  (
( F : NN --> H  /\  F  ~~>v  x )  <-> 
( F : NN --> H  /\  F  ~~>v  A ) ) )
16 eleq1 2539 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  H  <->  A  e.  H ) )
1715, 16imbi12d 320 . . . . . . 7  |-  ( x  =  A  ->  (
( ( F : NN
--> H  /\  F  ~~>v  x )  ->  x  e.  H )  <->  ( ( F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H )
) )
1813, 17spcv 3209 . . . . . 6  |-  ( A. x ( ( F : NN --> H  /\  F  ~~>v  x )  ->  x  e.  H )  ->  ( ( F : NN
--> H  /\  F  ~~>v  A )  ->  A  e.  H ) )
1912, 18syl6 33 . . . . 5  |-  ( 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 )
) )
206, 19syl 16 . . . 4  |-  ( ( 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 ) ) )
2120pm2.43b 50 . . 3  |-  ( A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  ->  ( ( F : NN
--> H  /\  F  ~~>v  A )  ->  A  e.  H ) )
222, 21syl 16 . 2  |-  ( H  e.  CH  ->  (
( F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H
) )
23223impib 1194 1  |-  ( ( H  e.  CH  /\  F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767   _Vcvv 3118   class class class wbr 4453   -->wf 5590   NNcn 10548    ~~>v chli 25667   SHcsh 25668   CHcch 25669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-map 7434  df-nn 10549  df-ch 25962
This theorem is referenced by:  hhsscms  26018  chintcli  26072  chscllem4  26381
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