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Theorem nmcopexi 22437
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
StepHypRef Expression
1 nmcopex.2 . . . 4  |-  T  e. 
ConOp
2 ax-hv0cl 21413 . . . 4  |-  0h  e.  ~H
3 1rp 10237 . . . 4  |-  1  e.  RR+
4 cnopc 22323 . . . 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 1282 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1 )
6 hvsub0 21485 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5381 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 3930 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcopex.1 . . . . . . . . . . 11  |-  T  e. 
LinOp
109lnop0i 22380 . . . . . . . . . 10  |-  ( T `
 0h )  =  0h
1110oveq2i 5721 . . . . . . . . 9  |-  ( ( T `  z )  -h  ( T `  0h ) )  =  ( ( T `  z
)  -h  0h )
129lnopfi 22379 . . . . . . . . . . 11  |-  T : ~H
--> ~H
1312ffvelrni 5516 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  ~H )
14 hvsub0 21485 . . . . . . . . . 10  |-  ( ( T `  z )  e.  ~H  ->  (
( T `  z
)  -h  0h )  =  ( T `  z ) )
1513, 14syl 17 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  0h )  =  ( T `  z ) )
1611, 15syl5eq 2297 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -h  ( T `
 0h ) )  =  ( T `  z ) )
1716fveq2d 5381 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  =  ( normh `  ( T `  z ) ) )
1817breq1d 3930 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( ( T `  z )  -h  ( T `  0h ) ) )  <  1  <->  ( normh `  ( T `  z )
)  <  1 ) )
198, 18imbi12d 313 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  (
normh `  ( ( T `
 z )  -h  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( normh `  ( T `  z
) )  <  1
) ) )
2019ralbiia 2537 . . . 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 2532 . . 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 201 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( normh `  ( T `  z ) )  <  1 )
23 nmopval 22266 . . 3  |-  ( T : ~H --> ~H  ->  (
normop `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( normh `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2412, 23ax-mp 10 . 2  |-  ( normop `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( normh `  ( T `  x ) ) ) } ,  RR* ,  <  )
2512ffvelrni 5516 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  ~H )
26 normcl 21534 . . 3  |-  ( ( T `  x )  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2725, 26syl 17 . 2  |-  ( x  e.  ~H  ->  ( normh `  ( T `  x ) )  e.  RR )
2810fveq2i 5380 . . 3  |-  ( normh `  ( T `  0h ) )  =  (
normh `  0h )
29 norm0 21537 . . 3  |-  ( normh `  0h )  =  0
3028, 29eqtri 2273 . 2  |-  ( normh `  ( T `  0h ) )  =  0
31 rpcn 10241 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
329lnopmuli 22382 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x ) ) )
3331, 32sylan 459 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  .h  ( T `  x
) ) )
3433fveq2d 5381 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( normh `  ( (
y  /  2 )  .h  ( T `  x ) ) ) )
35 norm-iii 21549 . . . 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 465 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( normh `  ( ( y  /  2 )  .h  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) ) )
37 rpre 10239 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
38 rpge0 10245 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3937, 38absidd 11782 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
4039adantr 453 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
4140oveq1d 5725 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( normh `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( normh `  ( T `  x ) ) ) )
4234, 36, 413eqtrrd 2290 . 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 22436 1  |-  ( normop `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   {cab 2239   A.wral 2509   E.wrex 2510   class class class wbr 3920   -->wf 4588   ` cfv 4592  (class class class)co 5710   supcsup 7077   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622   RR*cxr 8746    < clt 8747    <_ cle 8748    / cdiv 9303   2c2 9675   RR+crp 10233   abscabs 11596   ~Hchil 21329    .h csm 21331   normhcno 21333   0hc0v 21334    -h cmv 21335   normopcnop 21355   ConOpccop 21356   LinOpclo 21357
This theorem is referenced by:  nmcoplbi  22438  nmcopex  22439  cnlnadjlem2  22478  cnlnadjlem7  22483  cnlnadjlem8  22484
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hilex 21409  ax-hfvadd 21410  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-hnorm 21378  df-hvsub 21381  df-nmop 22249  df-cnop 22250  df-lnop 22251
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