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Theorem nmcopexi 23518
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 22494 . . . 4  |-  0h  e.  ~H
3 1rp 10605 . . . 4  |-  1  e.  RR+
4 cnopc 23404 . . . 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 1279 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 )
6 hvsub0 22566 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5723 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4214 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcopex.1 . . . . . . . . . . 11  |-  T  e. 
LinOp
109lnop0i 23461 . . . . . . . . . 10  |-  ( T `
 0h )  =  0h
1110oveq2i 6083 . . . . . . . . 9  |-  ( ( T `  z )  -h  ( T `  0h ) )  =  ( ( T `  z
)  -h  0h )
129lnopfi 23460 . . . . . . . . . . 11  |-  T : ~H
--> ~H
1312ffvelrni 5860 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  ~H )
14 hvsub0 22566 . . . . . . . . . 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 2479 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  ( T `
 0h ) )  =  ( T `  z ) )
1716fveq2d 5723 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  =  ( normh `  ( T `  z ) ) )
1817breq1d 4214 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1  <->  ( normh `  ( T `  z )
)  <  1 ) )
198, 18imbi12d 312 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  (
normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( normh `  ( T `  z
) )  <  1
) ) )
2019ralbiia 2729 . . . 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 2722 . . 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 200 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( normh `  ( T `  z ) )  <  1 )
23 nmopval 23347 . . 3  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( normh `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2412, 23ax-mp 8 . 2  |-  ( normop `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( normh `  ( T `  x ) ) ) } ,  RR* ,  <  )
2512ffvelrni 5860 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
26 normcl 22615 . . 3  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2725, 26syl 16 . 2  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2810fveq2i 5722 . . 3  |-  ( normh `  ( T `  0h ) )  =  (
normh `  0h )
29 norm0 22618 . . 3  |-  ( normh `  0h )  =  0
3028, 29eqtri 2455 . 2  |-  ( normh `  ( T `  0h ) )  =  0
31 rpcn 10609 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
329lnopmuli 23463 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x ) ) )
3331, 32sylan 458 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x
) ) )
3433fveq2d 5723 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( normh `  ( (
y  /  2 )  .h  ( T `  x ) ) ) )
35 norm-iii 22630 . . . 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 464 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( ( y  /  2 )  .h  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) ) )
37 rpre 10607 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
38 rpge0 10613 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3937, 38absidd 12213 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
4039adantr 452 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
4140oveq1d 6087 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( normh `  ( T `  x ) ) ) )
4234, 36, 413eqtrrd 2472 . 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 23517 1  |-  ( normop `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   class class class wbr 4204   -->wf 5441   ` cfv 5445  (class class class)co 6072   supcsup 7436   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984   RR*cxr 9108    < clt 9109    <_ cle 9110    / cdiv 9666   2c2 10038   RR+crp 10601   abscabs 12027   ~Hchil 22410    .h csm 22412   normhcno 22414   0hc0v 22415    -h cmv 22416   normopcnop 22436   ConOpccop 22437   LinOpclo 22438
This theorem is referenced by:  nmcoplbi  23519  nmcopex  23520  cnlnadjlem2  23559  cnlnadjlem7  23564  cnlnadjlem8  23565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-hilex 22490  ax-hfvadd 22491  ax-hvass 22493  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvdistr2 22500  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his3 22574  ax-his4 22575
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-hnorm 22459  df-hvsub 22462  df-nmop 23330  df-cnop 23331  df-lnop 23332
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