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Theorem hhcno 22598
Description: The continuous operators 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 )
Assertion
Ref Expression
hhcno  |-  ConOp  =  ( J  Cn  J )

Proof of Theorem hhcno
Dummy variables  x  w  y  z  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2628 . 2  |-  { t  e.  ( ~H  ^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) }  =  { t  |  ( t  e.  ( ~H 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) }
2 df-cnop 22534 . 2  |-  ConOp  =  {
t  e.  ( ~H 
^m  ~H )  |  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) }
3 hhcn.1 . . . . . . . . . . . . . 14  |-  D  =  ( normh  o.  -h  )
43hilmetdval 21889 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 21836 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2390 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( w  -h  x
) ) )
76adantll 694 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( x D w )  =  (
normh `  ( w  -h  x ) ) )
87breq1d 4114 . . . . . . . . . 10  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5746 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( t `  x
)  e.  ~H )
10 ffvelrn 5746 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  w  e.  ~H )  ->  ( t `  w
)  e.  ~H )
119, 10anim12dan 810 . . . . . . . . . . . . 13  |-  ( ( t : ~H --> ~H  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )  e.  ~H  /\  ( t `
 w )  e. 
~H ) )
123hilmetdval 21889 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( ( t `  x ) D ( t `  w ) )  =  ( normh `  ( ( t `  x )  -h  (
t `  w )
) ) )
13 normsub 21836 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( normh `  ( (
t `  x )  -h  ( t `  w
) ) )  =  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) ) )
1412, 13eqtrd 2390 . . . . . . . . . . . . 13  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( ( t `  x ) D ( t `  w ) )  =  ( normh `  ( ( t `  w )  -h  (
t `  x )
) ) )
1511, 14syl 15 . . . . . . . . . . . 12  |-  ( ( t : ~H --> ~H  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x ) D ( t `  w ) )  =  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) ) )
1615anassrs 629 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( t `
 x ) D ( t `  w
) )  =  (
normh `  ( ( t `
 w )  -h  ( t `  x
) ) ) )
1716breq1d 4114 . . . . . . . . . 10  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( ( t `  x ) D ( t `  w ) )  < 
y  <->  ( normh `  (
( t `  w
)  -h  ( t `
 x ) ) )  <  y ) )
188, 17imbi12d 311 . . . . . . . . 9  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( ( x D w )  <  z  ->  (
( t `  x
) D ( t `
 w ) )  <  y )  <->  ( ( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( t `  w )  -h  (
t `  x )
) )  <  y
) ) )
1918ralbidva 2635 . . . . . . . 8  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x ) D ( t `  w ) )  < 
y )  <->  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) )
2019rexbidv 2640 . . . . . . 7  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( E. z  e.  RR+  A. w  e.  ~H  ( ( x D w )  <  z  ->  ( ( t `  x ) D ( t `  w ) )  <  y )  <->  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) )
2120ralbidv 2639 . . . . . 6  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( x D w )  <  z  -> 
( ( t `  x ) D ( t `  w ) )  <  y )  <->  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) )
2221ralbidva 2635 . . . . 5  |-  ( t : ~H --> ~H  ->  ( A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x ) D ( t `  w ) )  < 
y )  <->  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( t `  w )  -h  (
t `  x )
) )  <  y
) ) )
2322pm5.32i 618 . . . 4  |-  ( ( t : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e. 
~H  ( ( x D w )  < 
z  ->  ( (
t `  x ) D ( t `  w ) )  < 
y ) )  <->  ( t : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( t `  w )  -h  (
t `  x )
) )  <  y
) ) )
243hilxmet 21888 . . . . 5  |-  D  e.  ( * Met `  ~H )
25 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
2625, 25metcn 18191 . . . . 5  |-  ( ( D  e.  ( * Met `  ~H )  /\  D  e.  ( * Met `  ~H )
)  ->  ( t  e.  ( J  Cn  J
)  <->  ( t : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( x D w )  <  z  -> 
( ( t `  x ) D ( t `  w ) )  <  y ) ) ) )
2724, 24, 26mp2an 653 . . . 4  |-  ( t  e.  ( J  Cn  J )  <->  ( t : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( x D w )  <  z  -> 
( ( t `  x ) D ( t `  w ) )  <  y ) ) )
28 ax-hilex 21693 . . . . . 6  |-  ~H  e.  _V
2928, 28elmap 6884 . . . . 5  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
3029anbi1i 676 . . . 4  |-  ( ( t  e.  ( ~H 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) )  <->  ( t : ~H --> ~H  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( normh `  ( ( t `  w )  -h  (
t `  x )
) )  <  y
) ) )
3123, 27, 303bitr4i 268 . . 3  |-  ( t  e.  ( J  Cn  J )  <->  ( t  e.  ( ~H  ^m  ~H )  /\  A. x  e. 
~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) )
3231abbi2i 2469 . 2  |-  ( J  Cn  J )  =  { t  |  ( t  e.  ( ~H 
^m  ~H )  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( normh `  ( (
t `  w )  -h  ( t `  x
) ) )  < 
y ) ) }
331, 2, 323eqtr4i 2388 1  |-  ConOp  =  ( J  Cn  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   E.wrex 2620   {crab 2623   class class class wbr 4104    o. ccom 4775   -->wf 5333   ` cfv 5337  (class class class)co 5945    ^m cmap 6860    < clt 8957   RR+crp 10446   * Metcxmt 16468   MetOpencmopn 16473    Cn ccn 17060   ~Hchil 21613   normhcno 21617    -h cmv 21619   ConOpccop 21640
This theorem is referenced by:  hmopidmchi  22845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907  ax-hilex 21693  ax-hfvadd 21694  ax-hvcom 21695  ax-hvass 21696  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvmulass 21701  ax-hvdistr1 21702  ax-hvdistr2 21703  ax-hvmul0 21704  ax-hfi 21772  ax-his1 21775  ax-his2 21776  ax-his3 21777  ax-his4 21778
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-n0 10058  df-z 10117  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-topgen 13443  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-top 16742  df-bases 16744  df-topon 16745  df-cn 17063  df-cnp 17064  df-grpo 20970  df-gid 20971  df-ginv 20972  df-gdiv 20973  df-ablo 21061  df-vc 21216  df-nv 21262  df-va 21265  df-ba 21266  df-sm 21267  df-0v 21268  df-vs 21269  df-nmcv 21270  df-ims 21271  df-hnorm 21662  df-hvsub 21665  df-cnop 22534
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