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Theorem hhcnf 22485
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 2552 . 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 22427 . 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 21775 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 21722 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( w  -h  x
) ) )
76adantll 694 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( x D w )  =  (
normh `  ( w  -h  x ) ) )
87breq1d 4033 . . . . . . . . . 10  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( t `  x
)  e.  CC )
10 ffvelrn 5663 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  w  e.  ~H )  ->  ( t `  w
)  e.  CC )
119, 10anim12dan 810 . . . . . . . . . . . . 13  |-  ( ( t : ~H --> CC  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )  e.  CC  /\  ( t `
 w )  e.  CC ) )
12 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1312cnmetdval 18280 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  x )  -  ( t `  w ) ) ) )
14 abssub 11810 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( abs `  (
( t `  x
)  -  ( t `
 w ) ) )  =  ( abs `  ( ( t `  w )  -  (
t `  x )
) ) )
1513, 14eqtrd 2315 . . . . . . . . . . . . 13  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1611, 15syl 15 . . . . . . . . . . . 12  |-  ( ( t : ~H --> CC  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )
( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1716anassrs 629 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  w )  -  ( t `  x ) ) ) )
1817breq1d 4033 . . . . . . . . . 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 311 . . . . . . . . 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 2559 . . . . . . . 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 2564 . . . . . . 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 2563 . . . . . 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 2559 . . . . 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 618 . . . 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 21774 . . . . 5  |-  D  e.  ( * Met `  ~H )
26 cnxmet 18282 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
27 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
28 hhcn.4 . . . . . . 7  |-  K  =  ( TopOpen ` fld )
2928cnfldtopn 18291 . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
3027, 29metcn 18089 . . . . 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 653 . . . 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 8818 . . . . . 6  |-  CC  e.  _V
33 ax-hilex 21579 . . . . . 6  |-  ~H  e.  _V
3432, 33elmap 6796 . . . . 5  |-  ( t  e.  ( CC  ^m  ~H )  <->  t : ~H --> CC )
3534anbi1i 676 . . . 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 268 . . 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 2394 . 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 2313 1  |-  ConFn  =  ( J  Cn  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547   class class class wbr 4023    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735    < clt 8867    - cmin 9037   RR+crp 10354   abscabs 11719   TopOpenctopn 13326   * Metcxmt 16369   MetOpencmopn 16372  ℂfldccnfld 16377    Cn ccn 16954   ~Hchil 21499   normhcno 21503    -h cmv 21505   ConFnccnfn 21533
This theorem is referenced by:  nlelchi  22641
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-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  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-int 3863  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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cnp 16958  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-hnorm 21548  df-hvsub 21551  df-cnfn 22427
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