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Theorem nmcfnexi 23507
Description: The norm of a continuous linear Hilbert space functional exists. Theorem 3.5(i) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (Proof shortened by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcfnex.1  |-  T  e. 
LinFn
nmcfnex.2  |-  T  e. 
ConFn
Assertion
Ref Expression
nmcfnexi  |-  ( normfn `  T )  e.  RR

Proof of Theorem nmcfnexi
Dummy variables  x  m  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmcfnex.2 . . . 4  |-  T  e. 
ConFn
2 ax-hv0cl 22459 . . . 4  |-  0h  e.  ~H
3 1rp 10572 . . . 4  |-  1  e.  RR+
4 cnfnc 23386 . . . 4  |-  ( ( T  e.  ConFn  /\  0h  e.  ~H  /\  1  e.  RR+ )  ->  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1 ) )
51, 2, 3, 4mp3an 1279 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1 )
6 hvsub0 22531 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5691 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4182 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcfnex.1 . . . . . . . . . . 11  |-  T  e. 
LinFn
109lnfn0i 23498 . . . . . . . . . 10  |-  ( T `
 0h )  =  0
1110oveq2i 6051 . . . . . . . . 9  |-  ( ( T `  z )  -  ( T `  0h ) )  =  ( ( T `  z
)  -  0 )
129lnfnfi 23497 . . . . . . . . . . 11  |-  T : ~H
--> CC
1312ffvelrni 5828 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  CC )
1413subid1d 9356 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  0 )  =  ( T `  z ) )
1511, 14syl5eq 2448 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  ( T `
 0h ) )  =  ( T `  z ) )
1615fveq2d 5691 . . . . . . 7  |-  ( z  e.  ~H  ->  ( abs `  ( ( T `
 z )  -  ( T `  0h )
) )  =  ( abs `  ( T `
 z ) ) )
1716breq1d 4182 . . . . . 6  |-  ( z  e.  ~H  ->  (
( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1  <->  ( abs `  ( T `  z ) )  <  1 ) )
188, 17imbi12d 312 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  ( abs `  ( ( T `  z )  -  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( abs `  ( T `  z
) )  <  1
) ) )
1918ralbiia 2698 . . . 4  |-  ( A. z  e.  ~H  (
( normh `  ( z  -h  0h ) )  < 
y  ->  ( abs `  ( ( T `  z )  -  ( T `  0h )
) )  <  1
)  <->  A. z  e.  ~H  ( ( normh `  z
)  <  y  ->  ( abs `  ( T `
 z ) )  <  1 ) )
2019rexbii 2691 . . 3  |-  ( E. y  e.  RR+  A. z  e.  ~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1 )  <->  E. y  e.  RR+  A. z  e.  ~H  ( ( normh `  z )  <  y  ->  ( abs `  ( T `  z )
)  <  1 ) )
215, 20mpbi 200 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( abs `  ( T `  z )
)  <  1 )
22 nmfnval 23332 . . 3  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( abs `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2312, 22ax-mp 8 . 2  |-  ( normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( abs `  ( T `  x )
) ) } ,  RR* ,  <  )
2412ffvelrni 5828 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  CC )
2524abscld 12193 . 2  |-  ( x  e.  ~H  ->  ( abs `  ( T `  x ) )  e.  RR )
2610fveq2i 5690 . . 3  |-  ( abs `  ( T `  0h ) )  =  ( abs `  0 )
27 abs0 12045 . . 3  |-  ( abs `  0 )  =  0
2826, 27eqtri 2424 . 2  |-  ( abs `  ( T `  0h ) )  =  0
29 rpcn 10576 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
309lnfnmuli 23500 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x ) ) )
3129, 30sylan 458 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x
) ) )
3231fveq2d 5691 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) ) )
33 absmul 12054 . . . 4  |-  ( ( ( y  /  2
)  e.  CC  /\  ( T `  x )  e.  CC )  -> 
( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) )  =  ( ( abs `  ( y  /  2 ) )  x.  ( abs `  ( T `  x )
) ) )
3429, 24, 33syl2an 464 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( ( y  /  2 )  x.  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) ) )
35 rpre 10574 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
36 rpge0 10580 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3735, 36absidd 12180 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
3837adantr 452 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
3938oveq1d 6055 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( abs `  ( T `  x ) ) ) )
4032, 34, 393eqtrrd 2441 . 2  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( y  /  2
)  x.  ( abs `  ( T `  x
) ) )  =  ( abs `  ( T `  ( (
y  /  2 )  .h  x ) ) ) )
4121, 23, 25, 28, 40nmcexi 23482 1  |-  ( normfn `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   class class class wbr 4172   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   RR+crp 10568   abscabs 11994   ~Hchil 22375    .h csm 22377   normhcno 22379   0hc0v 22380    -h cmv 22381   normfncnmf 22407   ConFnccnfn 22409   LinFnclf 22410
This theorem is referenced by:  nmcfnlbi  23508  nmcfnex  23509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-hilex 22455  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-hnorm 22424  df-hvsub 22427  df-nmfn 23301  df-cnfn 23303  df-lnfn 23304
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