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Theorem hhcnf 23408
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 2714 . 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 23350 . 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 22698 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 22645 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2468 . . . . . . . . . . . 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 4222 . . . . . . . . . 10  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5868 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( t `  x
)  e.  CC )
10 ffvelrn 5868 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1312cnmetdval 18805 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  x )  -  ( t `  w ) ) ) )
14 abssub 12130 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( abs `  (
( t `  x
)  -  ( t `
 w ) ) )  =  ( abs `  ( ( t `  w )  -  (
t `  x )
) ) )
1513, 14eqtrd 2468 . . . . . . . . . . . . 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 4222 . . . . . . . . . 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 2721 . . . . . . . 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 2726 . . . . . . 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 2725 . . . . . 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 2721 . . . . 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 22697 . . . . 5  |-  D  e.  ( * Met `  ~H )
26 cnxmet 18807 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
27 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
28 hhcn.4 . . . . . . 7  |-  K  =  ( TopOpen ` fld )
2928cnfldtopn 18816 . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
3027, 29metcn 18573 . . . . 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 9071 . . . . . 6  |-  CC  e.  _V
33 ax-hilex 22502 . . . . . 6  |-  ~H  e.  _V
3432, 33elmap 7042 . . . . 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 2547 . 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 2466 1  |-  ConFn  =  ( J  Cn  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705   E.wrex 2706   {crab 2709   class class class wbr 4212    o. ccom 4882   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   CCcc 8988    < clt 9120    - cmin 9291   RR+crp 10612   abscabs 12039   TopOpenctopn 13649   * Metcxmt 16686   MetOpencmopn 16691  ℂfldccnfld 16703    Cn ccn 17288   ~Hchil 22422   normhcno 22426    -h cmv 22428   ConFnccnfn 22456
This theorem is referenced by:  nlelchi  23564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070  ax-hilex 22502  ax-hfvadd 22503  ax-hvcom 22504  ax-hvass 22505  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmulass 22510  ax-hvdistr1 22511  ax-hvdistr2 22512  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his2 22585  ax-his3 22586  ax-his4 22587
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-rp 10613  df-xneg 10710  df-xadd 10711  df-xmul 10712  df-fz 11044  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-rest 13650  df-topn 13651  df-topgen 13667  df-psmet 16694  df-xmet 16695  df-met 16696  df-bl 16697  df-mopn 16698  df-cnfld 16704  df-top 16963  df-bases 16965  df-topon 16966  df-cn 17291  df-cnp 17292  df-grpo 21779  df-gid 21780  df-ginv 21781  df-gdiv 21782  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-vs 22078  df-nmcv 22079  df-ims 22080  df-hnorm 22471  df-hvsub 22474  df-cnfn 23350
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