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Theorem nmcfnexi 22624
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
Dummy variables  x  m  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem nmcfnexi
StepHypRef Expression
1 nmcfnex.2 . . . 4  |-  T  e. 
ConFn
2 ax-hv0cl 21576 . . . 4  |-  0h  e.  ~H
3 1rp 10354 . . . 4  |-  1  e.  RR+
4 cnfnc 22503 . . . 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 21648 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5490 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4035 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcfnex.1 . . . . . . . . . . 11  |-  T  e. 
LinFn
109lnfn0i 22615 . . . . . . . . . 10  |-  ( T `
 0h )  =  0
1110oveq2i 5831 . . . . . . . . 9  |-  ( ( T `  z )  -  ( T `  0h ) )  =  ( ( T `  z
)  -  0 )
129lnfnfi 22614 . . . . . . . . . . 11  |-  T : ~H
--> CC
1312ffvelrni 5626 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  CC )
1413subid1d 9142 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  0 )  =  ( T `  z ) )
1511, 14syl5eq 2329 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  ( T `
 0h ) )  =  ( T `  z ) )
1615fveq2d 5490 . . . . . . 7  |-  ( z  e.  ~H  ->  ( abs `  ( ( T `
 z )  -  ( T `  0h )
) )  =  ( abs `  ( T `
 z ) ) )
1716breq1d 4035 . . . . . 6  |-  ( z  e.  ~H  ->  (
( abs `  (
( T `  z
)  -  ( T `
 0h ) ) )  <  1  <->  ( abs `  ( T `  z ) )  <  1 ) )
188, 17imbi12d 313 . . . . 5  |-  ( z  e.  ~H  ->  (
( ( normh `  (
z  -h  0h )
)  <  y  ->  ( abs `  ( ( T `  z )  -  ( T `  0h ) ) )  <  1 )  <->  ( ( normh `  z )  < 
y  ->  ( abs `  ( T `  z
) )  <  1
) ) )
1918ralbiia 2577 . . . 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 2570 . . 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 201 . 2  |-  E. y  e.  RR+  A. z  e. 
~H  ( ( normh `  z )  <  y  ->  ( abs `  ( T `  z )
)  <  1 )
22 nmfnval 22449 . . 3  |-  ( T : ~H --> CC  ->  (
normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  ( ( normh `  x
)  <_  1  /\  m  =  ( abs `  ( T `  x
) ) ) } ,  RR* ,  <  )
)
2312, 22ax-mp 10 . 2  |-  ( normfn `  T )  =  sup ( { m  |  E. x  e.  ~H  (
( normh `  x )  <_  1  /\  m  =  ( abs `  ( T `  x )
) ) } ,  RR* ,  <  )
2412ffvelrni 5626 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  CC )
2524abscld 11913 . 2  |-  ( x  e.  ~H  ->  ( abs `  ( T `  x ) )  e.  RR )
2610fveq2i 5489 . . 3  |-  ( abs `  ( T `  0h ) )  =  ( abs `  0 )
27 abs0 11765 . . 3  |-  ( abs `  0 )  =  0
2826, 27eqtri 2305 . 2  |-  ( abs `  ( T `  0h ) )  =  0
29 rpcn 10358 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
309lnfnmuli 22617 . . . . 5  |-  ( ( ( y  /  2
)  e.  CC  /\  x  e.  ~H )  ->  ( T `  (
( y  /  2
)  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x ) ) )
3129, 30sylan 459 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( T `  ( (
y  /  2 )  .h  x ) )  =  ( ( y  /  2 )  x.  ( T `  x
) ) )
3231fveq2d 5490 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) ) )
33 absmul 11774 . . . 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 465 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( ( y  /  2 )  x.  ( T `  x
) ) )  =  ( ( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) ) )
35 rpre 10356 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
36 rpge0 10362 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3735, 36absidd 11900 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( abs `  ( y  /  2
) )  =  ( y  /  2 ) )
3837adantr 453 . . . 4  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( y  / 
2 ) )  =  ( y  /  2
) )
3938oveq1d 5835 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( abs `  ( T `  x ) ) ) )
4032, 34, 393eqtrrd 2322 . 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 22599 1  |-  ( normfn `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2271   A.wral 2545   E.wrex 2546   class class class wbr 4025   -->wf 5218   ` cfv 5222  (class class class)co 5820   supcsup 7189   CCcc 8731   RRcr 8732   0cc0 8733   1c1 8734    x. cmul 8738   RR*cxr 8862    < clt 8863    <_ cle 8864    - cmin 9033    / cdiv 9419   2c2 9791   RR+crp 10350   abscabs 11714   ~Hchil 21492    .h csm 21494   normhcno 21496   0hc0v 21497    -h cmv 21498   normfncnmf 21524   ConFnccnfn 21526   LinFnclf 21527
This theorem is referenced by:  nmcfnlbi  22625  nmcfnex  22626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-hilex 21572  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvmulass 21580  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his3 21656  ax-his4 21657
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-hnorm 21541  df-hvsub 21544  df-nmfn 22418  df-cnfn 22420  df-lnfn 22421
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