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Theorem nmcopexi 26760
Description: The norm of a continuous linear Hilbert space operator exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 5-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcopex.1  |-  T  e. 
LinOp
nmcopex.2  |-  T  e. 
ConOp
Assertion
Ref Expression
nmcopexi  |-  ( normop `  T )  e.  RR

Proof of Theorem nmcopexi
Dummy variables  x  m  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmcopex.2 . . . 4  |-  T  e. 
ConOp
2 ax-hv0cl 25734 . . . 4  |-  0h  e.  ~H
3 1rp 11236 . . . 4  |-  1  e.  RR+
4 cnopc 26646 . . . 4  |-  ( ( T  e.  ConOp  /\  0h  e.  ~H  /\  1  e.  RR+ )  ->  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 ) )
51, 2, 3, 4mp3an 1324 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 )
6 hvsub0 25807 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5876 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4463 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcopex.1 . . . . . . . . . . 11  |-  T  e. 
LinOp
109lnop0i 26703 . . . . . . . . . 10  |-  ( T `
 0h )  =  0h
1110oveq2i 6306 . . . . . . . . 9  |-  ( ( T `  z )  -h  ( T `  0h ) )  =  ( ( T `  z
)  -h  0h )
129lnopfi 26702 . . . . . . . . . . 11  |-  T : ~H
--> ~H
1312ffvelrni 6031 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  ~H )
14 hvsub0 25807 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ~H  ->  (
( T `  z
)  -h  0h )  =  ( T `  z ) )
1513, 14syl 16 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  0h )  =  ( T `  z ) )
1611, 15syl5eq 2520 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  ( T `
 0h ) )  =  ( T `  z ) )
1716fveq2d 5876 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  =  ( normh `  ( T `  z ) ) )
1817breq1d 4463 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1  <->  ( normh `  ( T `  z )
)  <  1 ) )
198, 18imbi12d 320 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  (
normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( normh `  ( T `  z
) )  <  1
) ) )
2019ralbiia 2897 . . . 4  |-  ( A. z  e.  ~H  (
( normh `  ( z  -h  0h ) )  < 
y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h )
) )  <  1
)  <->  A. z  e.  ~H  ( ( normh `  z
)  <  y  ->  (
normh `  ( T `  z ) )  <  1 ) )
2120rexbii 2969 . . 3  |-  ( E. y  e.  RR+  A. z  e.  ~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 )  <->  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( normh `  ( T `  z ) )  <  1 ) )
225, 21mpbi 208 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( normh `  ( T `  z ) )  <  1 )
23 nmopval 26589 . . 3  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( normh `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2412, 23ax-mp 5 . 2  |-  ( normop `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( normh `  ( T `  x ) ) ) } ,  RR* ,  <  )
2512ffvelrni 6031 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
26 normcl 25856 . . 3  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2725, 26syl 16 . 2  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2810fveq2i 5875 . . 3  |-  ( normh `  ( T `  0h ) )  =  (
normh `  0h )
29 norm0 25859 . . 3  |-  ( normh `  0h )  =  0
3028, 29eqtri 2496 . 2  |-  ( normh `  ( T `  0h ) )  =  0
31 rpcn 11240 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
329lnopmuli 26705 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x ) ) )
3331, 32sylan 471 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x
) ) )
3433fveq2d 5876 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( normh `  ( (
y  /  2 )  .h  ( T `  x ) ) ) )
35 norm-iii 25871 . . . 4  |-  ( ( ( y  /  2
)  e.  CC  /\  ( T `  x )  e.  ~H )  -> 
( normh `  ( (
y  /  2 )  .h  ( T `  x ) ) )  =  ( ( abs `  ( y  /  2
) )  x.  ( normh `  ( T `  x ) ) ) )
3631, 25, 35syl2an 477 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( ( y  /  2 )  .h  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) ) )
37 rpre 11238 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
38 rpge0 11244 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3937, 38absidd 13234 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
4039adantr 465 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
4140oveq1d 6310 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( normh `  ( T `  x ) ) ) )
4234, 36, 413eqtrrd 2513 . 2  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( y  /  2
)  x.  ( normh `  ( T `  x
) ) )  =  ( normh `  ( T `  ( ( y  / 
2 )  .h  x
) ) ) )
4322, 24, 27, 30, 42nmcexi 26759 1  |-  ( normop `  T )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   E.wrex 2818   class class class wbr 4453   -->wf 5590   ` cfv 5594  (class class class)co 6295   supcsup 7912   CCcc 9502   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509   RR*cxr 9639    < clt 9640    <_ cle 9641    / cdiv 10218   2c2 10597   RR+crp 11232   abscabs 13047   ~Hchil 25650    .h csm 25652   normhcno 25654   0hc0v 25655    -h cmv 25656   normopcnop 25676   ConOpccop 25677   LinOpclo 25678
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-hilex 25730  ax-hfvadd 25731  ax-hvass 25733  ax-hv0cl 25734  ax-hvaddid 25735  ax-hfvmul 25736  ax-hvmulid 25737  ax-hvmulass 25738  ax-hvdistr2 25740  ax-hvmul0 25741  ax-hfi 25810  ax-his1 25813  ax-his3 25815  ax-his4 25816
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-hnorm 25699  df-hvsub 25702  df-nmop 26572  df-cnop 26573  df-lnop 26574
This theorem is referenced by:  nmcoplbi  26761  nmcopex  26762  cnlnadjlem2  26801  cnlnadjlem7  26806  cnlnadjlem8  26807
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