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Theorem nmcfnexi 25460
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 24410 . . . 4  |-  0h  e.  ~H
3 1rp 11000 . . . 4  |-  1  e.  RR+
4 cnfnc 25339 . . . 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 1314 . . 3  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  ( z  -h  0h ) )  <  y  ->  ( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1 )
6 hvsub0 24483 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5700 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4307 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcfnex.1 . . . . . . . . . . 11  |-  T  e. 
LinFn
109lnfn0i 25451 . . . . . . . . . 10  |-  ( T `
 0h )  =  0
1110oveq2i 6107 . . . . . . . . 9  |-  ( ( T `  z )  -  ( T `  0h ) )  =  ( ( T `  z
)  -  0 )
129lnfnfi 25450 . . . . . . . . . . 11  |-  T : ~H
--> CC
1312ffvelrni 5847 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  CC )
1413subid1d 9713 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  0 )  =  ( T `  z ) )
1511, 14syl5eq 2487 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  ( T `
 0h ) )  =  ( T `  z ) )
1615fveq2d 5700 . . . . . . 7  |-  ( z  e.  ~H  ->  ( abs `  ( ( T `
 z )  -  ( T `  0h )
) )  =  ( abs `  ( T `
 z ) ) )
1716breq1d 4307 . . . . . 6  |-  ( z  e.  ~H  ->  (
( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1  <->  ( abs `  ( T `  z ) )  <  1 ) )
188, 17imbi12d 320 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  ( abs `  ( ( T `  z )  -  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( abs `  ( T `  z
) )  <  1
) ) )
1918ralbiia 2752 . . . 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 2745 . . 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 208 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( abs `  ( T `  z )
)  <  1 )
22 nmfnval 25285 . . 3  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( abs `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2312, 22ax-mp 5 . 2  |-  ( normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( abs `  ( T `  x )
) ) } ,  RR* ,  <  )
2412ffvelrni 5847 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  CC )
2524abscld 12927 . 2  |-  ( x  e.  ~H  ->  ( abs `  ( T `  x ) )  e.  RR )
2610fveq2i 5699 . . 3  |-  ( abs `  ( T `  0h ) )  =  ( abs `  0 )
27 abs0 12779 . . 3  |-  ( abs `  0 )  =  0
2826, 27eqtri 2463 . 2  |-  ( abs `  ( T `  0h ) )  =  0
29 rpcn 11004 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
309lnfnmuli 25453 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x ) ) )
3129, 30sylan 471 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x
) ) )
3231fveq2d 5700 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) ) )
33 absmul 12788 . . . 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 477 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( ( y  /  2 )  x.  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) ) )
35 rpre 11002 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
36 rpge0 11008 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3735, 36absidd 12914 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
3837adantr 465 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
3938oveq1d 6111 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( abs `  ( T `  x ) ) ) )
4032, 34, 393eqtrrd 2480 . 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 25435 1  |-  ( normfn `  T )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096   supcsup 7695   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292   RR*cxr 9422    < clt 9423    <_ cle 9424    - cmin 9600    / cdiv 9998   2c2 10376   RR+crp 10996   abscabs 12728   ~Hchil 24326    .h csm 24328   normhcno 24330   0hc0v 24331    -h cmv 24332   normfncnmf 24358   ConFnccnfn 24360   LinFnclf 24361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-hilex 24406  ax-hv0cl 24410  ax-hvaddid 24411  ax-hfvmul 24412  ax-hvmulid 24413  ax-hvmulass 24414  ax-hvmul0 24417  ax-hfi 24486  ax-his1 24489  ax-his3 24491  ax-his4 24492
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-hnorm 24375  df-hvsub 24378  df-nmfn 25254  df-cnfn 25256  df-lnfn 25257
This theorem is referenced by:  nmcfnlbi  25461  nmcfnex  25462
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