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Theorem hhcnf 22501
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 2565 . 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 22443 . 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 21791 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 21738 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2328 . . . . . . . . . . . 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 4049 . . . . . . . . . 10  |-  ( ( ( t : ~H --> CC  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> CC  /\  x  e.  ~H )  ->  ( t `  x
)  e.  CC )
10 ffvelrn 5679 . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . 15  |-  ( abs 
o.  -  )  =  ( abs  o.  -  )
1312cnmetdval 18296 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( ( t `
 x ) ( abs  o.  -  )
( t `  w
) )  =  ( abs `  ( ( t `  x )  -  ( t `  w ) ) ) )
14 abssub 11826 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  CC  /\  ( t `  w
)  e.  CC )  ->  ( abs `  (
( t `  x
)  -  ( t `
 w ) ) )  =  ( abs `  ( ( t `  w )  -  (
t `  x )
) ) )
1513, 14eqtrd 2328 . . . . . . . . . . . . 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 4049 . . . . . . . . . 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 2572 . . . . . . . 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 2577 . . . . . . 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 2576 . . . . . 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 2572 . . . . 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 21790 . . . . 5  |-  D  e.  ( * Met `  ~H )
26 cnxmet 18298 . . . . 5  |-  ( abs 
o.  -  )  e.  ( * Met `  CC )
27 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
28 hhcn.4 . . . . . . 7  |-  K  =  ( TopOpen ` fld )
2928cnfldtopn 18307 . . . . . 6  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
3027, 29metcn 18105 . . . . 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 8834 . . . . . 6  |-  CC  e.  _V
33 ax-hilex 21595 . . . . . 6  |-  ~H  e.  _V
3432, 33elmap 6812 . . . . 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 2407 . 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 2326 1  |-  ConFn  =  ( J  Cn  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560   class class class wbr 4039    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751    < clt 8883    - cmin 9053   RR+crp 10370   abscabs 11735   TopOpenctopn 13342   * Metcxmt 16385   MetOpencmopn 16388  ℂfldccnfld 16393    Cn ccn 16970   ~Hchil 21515   normhcno 21519    -h cmv 21521   ConFnccnfn 21549
This theorem is referenced by:  nlelchi  22657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-rest 13343  df-topn 13344  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-cnp 16974  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-hnorm 21564  df-hvsub 21567  df-cnfn 22443
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