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Theorem chlimi 21814
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 21803 . . . 4  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
21simprbi 450 . . 3  |-  ( H  e.  CH  ->  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
3 nnex 9752 . . . . . . 7  |-  NN  e.  _V
4 fex 5749 . . . . . . 7  |-  ( ( F : NN --> H  /\  NN  e.  _V )  ->  F  e.  _V )
53, 4mpan2 652 . . . . . 6  |-  ( F : NN --> H  ->  F  e.  _V )
65adantr 451 . . . . 5  |-  ( ( F : NN --> H  /\  F  ~~>v  A )  ->  F  e.  _V )
7 feq1 5375 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f : NN --> H  <->  F : NN
--> H ) )
8 breq1 4026 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f  ~~>v  x  <->  F  ~~>v  x ) )
97, 8anbi12d 691 . . . . . . . . 9  |-  ( f  =  F  ->  (
( f : NN --> H  /\  f  ~~>v  x )  <-> 
( F : NN --> H  /\  F  ~~>v  x ) ) )
109imbi1d 308 . . . . . . . 8  |-  ( f  =  F  ->  (
( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( ( F : NN --> H  /\  F  ~~>v  x )  ->  x  e.  H )
) )
1110albidv 1611 . . . . . . 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 2868 . . . . . 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 4027 . . . . . . . . 9  |-  ( x  =  A  ->  ( F  ~~>v  x  <->  F  ~~>v  A ) )
1514anbi2d 684 . . . . . . . 8  |-  ( x  =  A  ->  (
( F : NN --> H  /\  F  ~~>v  x )  <-> 
( F : NN --> H  /\  F  ~~>v  A ) ) )
16 eleq1 2343 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  H  <->  A  e.  H ) )
1715, 16imbi12d 311 . . . . . . 7  |-  ( x  =  A  ->  (
( ( F : NN
--> H  /\  F  ~~>v  x )  ->  x  e.  H )  <->  ( ( F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H )
) )
1813, 17spcv 2874 . . . . . 6  |-  ( A. x ( ( F : NN --> H  /\  F  ~~>v  x )  ->  x  e.  H )  ->  ( ( F : NN
--> H  /\  F  ~~>v  A )  ->  A  e.  H ) )
1912, 18syl6 29 . . . . 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 15 . . . 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 46 . . 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 15 . 2  |-  ( H  e.  CH  ->  (
( F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H
) )
23223impib 1149 1  |-  ( ( H  e.  CH  /\  F : NN --> H  /\  F  ~~>v  A )  ->  A  e.  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   -->wf 5251   NNcn 9746    ~~>v chli 21507   SHcsh 21508   CHcch 21509
This theorem is referenced by:  hhsscms  21856  chintcli  21910  chscllem4  22219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-map 6774  df-nn 9747  df-ch 21801
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