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Theorem chlimi 21869
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 21858 . . . 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 9797 . . . . . . 7  |-  NN  e.  _V
4 fex 5790 . . . . . . 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 5412 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f : NN --> H  <->  F : NN
--> H ) )
8 breq1 4063 . . . . . . . . . 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 1616 . . . . . . 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 2902 . . . . . 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 4064 . . . . . . . . 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 2376 . . . . . . . 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 2908 . . . . . 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 1531    = wceq 1633    e. wcel 1701   _Vcvv 2822   class class class wbr 4060   -->wf 5288   NNcn 9791    ~~>v chli 21562   SHcsh 21563   CHcch 21564
This theorem is referenced by:  hhsscms  21911  chintcli  21965  chscllem4  22274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-i2m1 8850  ax-1ne0 8851  ax-rrecex 8854  ax-cnre 8855
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-map 6817  df-nn 9792  df-ch 21856
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