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Theorem hhcno 26596
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 2823 . 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 26532 . 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 25886 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( x  -h  w
) ) )
5 normsub 25833 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( normh `  ( x  -h  w ) )  =  ( normh `  ( w  -h  x ) ) )
64, 5eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( x D w )  =  ( normh `  ( w  -h  x
) ) )
76adantll 713 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( x D w )  =  (
normh `  ( w  -h  x ) ) )
87breq1d 4457 . . . . . . . . . 10  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x D w )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  z ) )
9 ffvelrn 6020 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  x  e.  ~H )  ->  ( t `  x
)  e.  ~H )
10 ffvelrn 6020 . . . . . . . . . . . . . 14  |-  ( ( t : ~H --> ~H  /\  w  e.  ~H )  ->  ( t `  w
)  e.  ~H )
119, 10anim12dan 835 . . . . . . . . . . . . 13  |-  ( ( t : ~H --> ~H  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
t `  x )  e.  ~H  /\  ( t `
 w )  e. 
~H ) )
123hilmetdval 25886 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( ( t `  x ) D ( t `  w ) )  =  ( normh `  ( ( t `  x )  -h  (
t `  w )
) ) )
13 normsub 25833 . . . . . . . . . . . . . 14  |-  ( ( ( t `  x
)  e.  ~H  /\  ( t `  w
)  e.  ~H )  ->  ( normh `  ( (
t `  x )  -h  ( t `  w
) ) )  =  ( normh `  ( (
t `  w )  -h  ( t `  x
) ) ) )
1412, 13eqtrd 2508 . . . . . . . . . . . . 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 648 . . . . . . . . . . 11  |-  ( ( ( t : ~H --> ~H  /\  x  e.  ~H )  /\  w  e.  ~H )  ->  ( ( t `
 x ) D ( t `  w
) )  =  (
normh `  ( ( t `
 w )  -h  ( t `  x
) ) ) )
1716breq1d 4457 . . . . . . . . . 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 320 . . . . . . . . 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 2900 . . . . . . . 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 2973 . . . . . . 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 2903 . . . . . 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 2900 . . . . 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 637 . . . 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 25885 . . . . 5  |-  D  e.  ( *Met `  ~H )
25 hhcn.2 . . . . . 6  |-  J  =  ( MetOpen `  D )
2625, 25metcn 20873 . . . . 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 672 . . . 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 25689 . . . . . 6  |-  ~H  e.  _V
2928, 28elmap 7448 . . . . 5  |-  ( t  e.  ( ~H  ^m  ~H )  <->  t : ~H --> ~H )
3029anbi1i 695 . . . 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 277 . . 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 2600 . 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 2506 1  |-  ConOp  =  ( J  Cn  J )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   {crab 2818   class class class wbr 4447    o. ccom 5003   -->wf 5584   ` cfv 5588  (class class class)co 6285    ^m cmap 7421    < clt 9629   RR+crp 11221   *Metcxmt 18214   MetOpencmopn 18219    Cn ccn 19531   ~Hchil 25609   normhcno 25613    -h cmv 25615   ConOpccop 25636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571  ax-addf 9572  ax-mulf 9573  ax-hilex 25689  ax-hfvadd 25690  ax-hvcom 25691  ax-hvass 25692  ax-hv0cl 25693  ax-hvaddid 25694  ax-hfvmul 25695  ax-hvmulid 25696  ax-hvmulass 25697  ax-hvdistr1 25698  ax-hvdistr2 25699  ax-hvmul0 25700  ax-hfi 25769  ax-his1 25772  ax-his2 25773  ax-his3 25774  ax-his4 25775
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-n0 10797  df-z 10866  df-uz 11084  df-q 11184  df-rp 11222  df-xneg 11319  df-xadd 11320  df-xmul 11321  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-topgen 14702  df-psmet 18222  df-xmet 18223  df-met 18224  df-bl 18225  df-mopn 18226  df-top 19206  df-bases 19208  df-topon 19209  df-cn 19534  df-cnp 19535  df-grpo 24966  df-gid 24967  df-ginv 24968  df-gdiv 24969  df-ablo 25057  df-vc 25212  df-nv 25258  df-va 25261  df-ba 25262  df-sm 25263  df-0v 25264  df-vs 25265  df-nmcv 25266  df-ims 25267  df-hnorm 25658  df-hvsub 25661  df-cnop 26532
This theorem is referenced by:  hmopidmchi  26843
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