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Theorem nmcfnexi 22631
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 21583 . . . 4  |-  0h  e.  ~H
3 1rp 10358 . . . 4  |-  1  e.  RR+
4 cnfnc 22510 . . . 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 1277 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1 )
6 hvsub0 21655 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5529 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4033 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcfnex.1 . . . . . . . . . . 11  |-  T  e. 
LinFn
109lnfn0i 22622 . . . . . . . . . 10  |-  ( T `
 0h )  =  0
1110oveq2i 5869 . . . . . . . . 9  |-  ( ( T `  z )  -  ( T `  0h ) )  =  ( ( T `  z
)  -  0 )
129lnfnfi 22621 . . . . . . . . . . 11  |-  T : ~H
--> CC
1312ffvelrni 5664 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  CC )
1413subid1d 9146 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  0 )  =  ( T `  z ) )
1511, 14syl5eq 2327 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  ( T `
 0h ) )  =  ( T `  z ) )
1615fveq2d 5529 . . . . . . 7  |-  ( z  e.  ~H  ->  ( abs `  ( ( T `
 z )  -  ( T `  0h )
) )  =  ( abs `  ( T `
 z ) ) )
1716breq1d 4033 . . . . . 6  |-  ( z  e.  ~H  ->  (
( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1  <->  ( abs `  ( T `  z ) )  <  1 ) )
188, 17imbi12d 311 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  ( abs `  ( ( T `  z )  -  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( abs `  ( T `  z
) )  <  1
) ) )
1918ralbiia 2575 . . . 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 2568 . . 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 199 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( abs `  ( T `  z )
)  <  1 )
22 nmfnval 22456 . . 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 5664 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  CC )
2524abscld 11918 . 2  |-  ( x  e.  ~H  ->  ( abs `  ( T `  x ) )  e.  RR )
2610fveq2i 5528 . . 3  |-  ( abs `  ( T `  0h ) )  =  ( abs `  0 )
27 abs0 11770 . . 3  |-  ( abs `  0 )  =  0
2826, 27eqtri 2303 . 2  |-  ( abs `  ( T `  0h ) )  =  0
29 rpcn 10362 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
309lnfnmuli 22624 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x ) ) )
3129, 30sylan 457 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x
) ) )
3231fveq2d 5529 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) ) )
33 absmul 11779 . . . 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 463 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( ( y  /  2 )  x.  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) ) )
35 rpre 10360 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
36 rpge0 10366 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3735, 36absidd 11905 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
3837adantr 451 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
3938oveq1d 5873 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( abs `  ( T `  x ) ) ) )
4032, 34, 393eqtrrd 2320 . 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 22606 1  |-  ( normfn `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   RR+crp 10354   abscabs 11719   ~Hchil 21499    .h csm 21501   normhcno 21503   0hc0v 21504    -h cmv 21505   normfncnmf 21531   ConFnccnfn 21533   LinFnclf 21534
This theorem is referenced by:  nmcfnlbi  22632  nmcfnex  22633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-hilex 21579  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his3 21663  ax-his4 21664
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-hnorm 21548  df-hvsub 21551  df-nmfn 22425  df-cnfn 22427  df-lnfn 22428
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