HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hhcnf Unicode version

Theorem hhcnf 23256
Description: The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
hhcn.1  |-  D  =  ( normh  o.  -h  )
hhcn.2  |-  J  =  ( MetOpen `  D )
hhcn.4  |-  K  =  ( TopOpen ` fld )
Assertion
Ref Expression
hhcnf  |-  ConFn  =  ( J  Cn  K )

Proof of Theorem hhcnf
Dummy variables  x  w  y  z  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2658 . 2  |-  { t  e.  ( CC  ^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) }  =  { t  |  ( t  e.  ( CC  ^m  ~H )  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) }
2 df-cnfn 23198 . 2  |-  ConFn  =  {
t  e.  ( CC 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) }
3 hhcn.1 . . . . . . . . . . . . . 14  |-  D  =  ( normh  o.  -h  )
43hilmetdval 22546 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 22493 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2419 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( w  -h  x
) ) )
76adantll 695 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( x D w )  =  (
normh `  ( w  -h  x ) ) )
87breq1d 4163 . . . . . . . . . 10  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5807 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( t `  x
)  e.  CC )
10 ffvelrn 5807 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  w  e.  ~H )  ->  ( t `  w
)  e.  CC )
119, 10anim12dan 811 . . . . . . . . . . . . 13  |-  ( ( t : ~H --> CC  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )  e.  CC  /\  ( t `
 w )  e.  CC ) )
12 eqid 2387 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1312cnmetdval 18676 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  x )  -  ( t `  w ) ) ) )
14 abssub 12057 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( abs `  (
( t `  x
)  -  ( t `
 w ) ) )  =  ( abs `  ( ( t `  w )  -  (
t `  x )
) ) )
1513, 14eqtrd 2419 . . . . . . . . . . . . 13  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1611, 15syl 16 . . . . . . . . . . . 12  |-  ( ( t : ~H --> CC  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )
( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1716anassrs 630 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1817breq1d 4163 . . . . . . . . . 10  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( ( t `  x ) ( abs  o.  -  ) ( t `  w ) )  < 
y  <->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) )
198, 18imbi12d 312 . . . . . . . . 9  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( ( x D w )  <  z  ->  (
( t `  x
) ( abs  o.  -  ) ( t `
 w ) )  <  y )  <->  ( ( normh `  ( w  -h  x ) )  < 
z  ->  ( abs `  ( ( t `  w )  -  (
t `  x )
) )  <  y
) ) )
2019ralbidva 2665 . . . . . . . 8  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x )
( abs  o.  -  )
( t `  w
) )  <  y
)  <->  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
2120rexbidv 2670 . . . . . . 7  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( E. z  e.  RR+  A. w  e.  ~H  ( ( x D w )  <  z  ->  ( ( t `  x ) ( abs 
o.  -  ) (
t `  w )
)  <  y )  <->  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
2221ralbidv 2669 . . . . . 6  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( x D w )  <  z  -> 
( ( t `  x ) ( abs 
o.  -  ) (
t `  w )
)  <  y )  <->  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
2322ralbidva 2665 . . . . 5  |-  ( t : ~H --> CC  ->  ( A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x )
( abs  o.  -  )
( t `  w
) )  <  y
)  <->  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
2423pm5.32i 619 . . . 4  |-  ( ( t : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x )
( abs  o.  -  )
( t `  w
) )  <  y
) )  <->  ( t : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
253hilxmet 22545 . . . . 5  |-  D  e.  ( * Met `  ~H )
26 cnxmet 18678 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
27 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
28 hhcn.4 . . . . . . 7  |-  K  =  ( TopOpen ` fld )
2928cnfldtopn 18687 . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
3027, 29metcn 18463 . . . . 5  |-  ( ( D  e.  ( * Met `  ~H )  /\  ( abs  o.  -  )  e.  ( * Met `  CC ) )  ->  ( t  e.  ( J  Cn  K
)  <->  ( t : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( x D w )  <  z  ->  ( ( t `  x ) ( abs 
o.  -  ) (
t `  w )
)  <  y )
) ) )
3125, 26, 30mp2an 654 . . . 4  |-  ( t  e.  ( J  Cn  K )  <->  ( t : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( x D w )  <  z  ->  ( ( t `  x ) ( abs 
o.  -  ) (
t `  w )
)  <  y )
) )
32 cnex 9004 . . . . . 6  |-  CC  e.  _V
33 ax-hilex 22350 . . . . . 6  |-  ~H  e.  _V
3432, 33elmap 6978 . . . . 5  |-  ( t  e.  ( CC  ^m  ~H )  <->  t : ~H --> CC )
3534anbi1i 677 . . . 4  |-  ( ( t  e.  ( CC 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) )  <->  ( t : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
3624, 31, 353bitr4i 269 . . 3  |-  ( t  e.  ( J  Cn  K )  <->  ( t  e.  ( CC  ^m  ~H )  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) )
3736abbi2i 2498 . 2  |-  ( J  Cn  K )  =  { t  |  ( t  e.  ( CC 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( t `  w
)  -  ( t `
 x ) ) )  <  y ) ) }
381, 2, 373eqtr4i 2417 1  |-  ConFn  =  ( J  Cn  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   E.wrex 2650   {crab 2653   class class class wbr 4153    o. ccom 4822   -->wf 5390   ` cfv 5394  (class class class)co 6020    ^m cmap 6954   CCcc 8921    < clt 9053    - cmin 9223   RR+crp 10544   abscabs 11966   TopOpenctopn 13576   * Metcxmt 16612   MetOpencmopn 16617  ℂfldccnfld 16626    Cn ccn 17210   ~Hchil 22270   normhcno 22274    -h cmv 22276   ConFnccnfn 22304
This theorem is referenced by:  nlelchi  23412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003  ax-hilex 22350  ax-hfvadd 22351  ax-hvcom 22352  ax-hvass 22353  ax-hv0cl 22354  ax-hvaddid 22355  ax-hfvmul 22356  ax-hvmulid 22357  ax-hvmulass 22358  ax-hvdistr1 22359  ax-hvdistr2 22360  ax-hvmul0 22361  ax-hfi 22429  ax-his1 22432  ax-his2 22433  ax-his3 22434  ax-his4 22435
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-fz 10976  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-rest 13577  df-topn 13578  df-topgen 13594  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-cn 17213  df-cnp 17214  df-grpo 21627  df-gid 21628  df-ginv 21629  df-gdiv 21630  df-ablo 21718  df-vc 21873  df-nv 21919  df-va 21922  df-ba 21923  df-sm 21924  df-0v 21925  df-vs 21926  df-nmcv 21927  df-ims 21928  df-hnorm 22319  df-hvsub 22322  df-cnfn 23198
  Copyright terms: Public domain W3C validator