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Theorem nmcopexi 27144
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 26118 . . . 4  |-  0h  e.  ~H
3 1rp 11225 . . . 4  |-  1  e.  RR+
4 cnopc 27030 . . . 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 1322 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 )
6 hvsub0 26191 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5852 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4449 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcopex.1 . . . . . . . . . . 11  |-  T  e. 
LinOp
109lnop0i 27087 . . . . . . . . . 10  |-  ( T `
 0h )  =  0h
1110oveq2i 6281 . . . . . . . . 9  |-  ( ( T `  z )  -h  ( T `  0h ) )  =  ( ( T `  z
)  -h  0h )
129lnopfi 27086 . . . . . . . . . . 11  |-  T : ~H
--> ~H
1312ffvelrni 6006 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  ~H )
14 hvsub0 26191 . . . . . . . . . 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 2507 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  ( T `
 0h ) )  =  ( T `  z ) )
1716fveq2d 5852 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  =  ( normh `  ( T `  z ) ) )
1817breq1d 4449 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1  <->  ( normh `  ( T `  z )
)  <  1 ) )
198, 18imbi12d 318 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  (
normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( normh `  ( T `  z
) )  <  1
) ) )
2019ralbiia 2884 . . . 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 2956 . . 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 26973 . . 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 6006 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
26 normcl 26240 . . 3  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2725, 26syl 16 . 2  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2810fveq2i 5851 . . 3  |-  ( normh `  ( T `  0h ) )  =  (
normh `  0h )
29 norm0 26243 . . 3  |-  ( normh `  0h )  =  0
3028, 29eqtri 2483 . 2  |-  ( normh `  ( T `  0h ) )  =  0
31 rpcn 11229 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
329lnopmuli 27089 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x ) ) )
3331, 32sylan 469 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x
) ) )
3433fveq2d 5852 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( normh `  ( (
y  /  2 )  .h  ( T `  x ) ) ) )
35 norm-iii 26255 . . . 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 475 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( ( y  /  2 )  .h  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) ) )
37 rpre 11227 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
38 rpge0 11233 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3937, 38absidd 13336 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
4039adantr 463 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
4140oveq1d 6285 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( normh `  ( T `  x ) ) ) )
4234, 36, 413eqtrrd 2500 . 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 27143 1  |-  ( normop `  T )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cab 2439   A.wral 2804   E.wrex 2805   class class class wbr 4439   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    / cdiv 10202   2c2 10581   RR+crp 11221   abscabs 13149   ~Hchil 26034    .h csm 26036   normhcno 26038   0hc0v 26039    -h cmv 26040   normopcnop 26060   ConOpccop 26061   LinOpclo 26062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-hilex 26114  ax-hfvadd 26115  ax-hvass 26117  ax-hv0cl 26118  ax-hvaddid 26119  ax-hfvmul 26120  ax-hvmulid 26121  ax-hvmulass 26122  ax-hvdistr2 26124  ax-hvmul0 26125  ax-hfi 26194  ax-his1 26197  ax-his3 26199  ax-his4 26200
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-seq 12090  df-exp 12149  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-hnorm 26083  df-hvsub 26086  df-nmop 26956  df-cnop 26957  df-lnop 26958
This theorem is referenced by:  nmcoplbi  27145  nmcopex  27146  cnlnadjlem2  27185  cnlnadjlem7  27190  cnlnadjlem8  27191
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