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Theorem hhcno 23360
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 2675 . 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 23296 . 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 22651 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 22598 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( w  -h  x
) ) )
76adantll 695 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( x D w )  =  (
normh `  ( w  -h  x ) ) )
87breq1d 4182 . . . . . . . . . 10  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( t `  x
)  e.  ~H )
10 ffvelrn 5827 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  w  e.  ~H )  ->  ( t `  w
)  e.  ~H )
119, 10anim12dan 811 . . . . . . . . . . . . 13  |-  ( ( t : ~H --> ~H  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )  e.  ~H  /\  ( t `
 w )  e. 
~H ) )
123hilmetdval 22651 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( ( t `  x ) D ( t `  w ) )  =  ( normh `  ( ( t `  x )  -h  (
t `  w )
) ) )
13 normsub 22598 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( normh `  ( (
t `  x )  -h  ( t `  w
) ) )  =  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) ) )
1412, 13eqtrd 2436 . . . . . . . . . . . . 13  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( ( t `  x ) D ( t `  w ) )  =  ( normh `  ( ( t `  w )  -h  (
t `  x )
) ) )
1511, 14syl 16 . . . . . . . . . . . 12  |-  ( ( t : ~H --> ~H  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x ) D ( t `  w ) )  =  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) ) )
1615anassrs 630 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( t `
 x ) D ( t `  w
) )  =  (
normh `  ( ( t `
 w )  -h  ( t `  x
) ) ) )
1716breq1d 4182 . . . . . . . . . 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 312 . . . . . . . . 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 2682 . . . . . . . 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 2687 . . . . . . 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 2686 . . . . . 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 2682 . . . . 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 619 . . . 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 22650 . . . . 5  |-  D  e.  ( * Met `  ~H )
25 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
2625, 25metcn 18526 . . . . 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 654 . . . 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 22455 . . . . . 6  |-  ~H  e.  _V
2928, 28elmap 7001 . . . . 5  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
3029anbi1i 677 . . . 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 269 . . 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 2515 . 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 2434 1  |-  ConOp  =  ( J  Cn  J )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   {crab 2670   class class class wbr 4172    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977    < clt 9076   RR+crp 10568   * Metcxmt 16641   MetOpencmopn 16646    Cn ccn 17242   ~Hchil 22375   normhcno 22379    -h cmv 22381   ConOpccop 22402
This theorem is referenced by:  hmopidmchi  23607
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245  df-cnp 17246  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-hnorm 22424  df-hvsub 22427  df-cnop 23296
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