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Theorem nmcfnexi 23537
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 22489 . . . 4  |-  0h  e.  ~H
3 1rp 10600 . . . 4  |-  1  e.  RR+
4 cnfnc 23416 . . . 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 22561 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
z  -h  0h )  =  z )
76fveq2d 5718 . . . . . . 7  |-  ( z  e.  ~H  ->  ( normh `  ( z  -h 
0h ) )  =  ( normh `  z )
)
87breq1d 4209 . . . . . 6  |-  ( z  e.  ~H  ->  (
( normh `  ( z  -h  0h ) )  < 
y  <->  ( normh `  z
)  <  y )
)
9 nmcfnex.1 . . . . . . . . . . 11  |-  T  e. 
LinFn
109lnfn0i 23528 . . . . . . . . . 10  |-  ( T `
 0h )  =  0
1110oveq2i 6078 . . . . . . . . 9  |-  ( ( T `  z )  -  ( T `  0h ) )  =  ( ( T `  z
)  -  0 )
129lnfnfi 23527 . . . . . . . . . . 11  |-  T : ~H
--> CC
1312ffvelrni 5855 . . . . . . . . . 10  |-  ( z  e.  ~H  ->  ( T `  z )  e.  CC )
1413subid1d 9384 . . . . . . . . 9  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  0 )  =  ( T `  z ) )
1511, 14syl5eq 2474 . . . . . . . 8  |-  ( z  e.  ~H  ->  (
( T `  z
)  -  ( T `
 0h ) )  =  ( T `  z ) )
1615fveq2d 5718 . . . . . . 7  |-  ( z  e.  ~H  ->  ( abs `  ( ( T `
 z )  -  ( T `  0h )
) )  =  ( abs `  ( T `
 z ) ) )
1716breq1d 4209 . . . . . 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 2724 . . . 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 2717 . . 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 23362 . . 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 5855 . . 3  |-  ( x  e.  ~H  ->  ( T `  x )  e.  CC )
2524abscld 12221 . 2  |-  ( x  e.  ~H  ->  ( abs `  ( T `  x ) )  e.  RR )
2610fveq2i 5717 . . 3  |-  ( abs `  ( T `  0h ) )  =  ( abs `  0 )
27 abs0 12073 . . 3  |-  ( abs `  0 )  =  0
2826, 27eqtri 2450 . 2  |-  ( abs `  ( T `  0h ) )  =  0
29 rpcn 10604 . . . . 5  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  CC )
309lnfnmuli 23530 . . . . 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 5718 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  ( abs `  ( T `  ( ( y  / 
2 )  .h  x
) ) )  =  ( abs `  (
( y  /  2
)  x.  ( T `
 x ) ) ) )
33 absmul 12082 . . . 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 10602 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  ( y  /  2 )  e.  RR )
36 rpge0 10608 . . . . . 6  |-  ( ( y  /  2 )  e.  RR+  ->  0  <_ 
( y  /  2
) )
3735, 36absidd 12208 . . . . 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 6082 . . 3  |-  ( ( ( y  /  2
)  e.  RR+  /\  x  e.  ~H )  ->  (
( abs `  (
y  /  2 ) )  x.  ( abs `  ( T `  x
) ) )  =  ( ( y  / 
2 )  x.  ( abs `  ( T `  x ) ) ) )
4032, 34, 393eqtrrd 2467 . 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 23512 1  |-  ( normfn `  T )  e.  RR
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2416   A.wral 2692   E.wrex 2693   class class class wbr 4199   -->wf 5436   ` cfv 5440  (class class class)co 6067   supcsup 7431   CCcc 8972   RRcr 8973   0cc0 8974   1c1 8975    x. cmul 8979   RR*cxr 9103    < clt 9104    <_ cle 9105    - cmin 9275    / cdiv 9661   2c2 10033   RR+crp 10596   abscabs 12022   ~Hchil 22405    .h csm 22407   normhcno 22409   0hc0v 22410    -h cmv 22411   normfncnmf 22437   ConFnccnfn 22439   LinFnclf 22440
This theorem is referenced by:  nmcfnlbi  23538  nmcfnex  23539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-1cn 9032  ax-icn 9033  ax-addcl 9034  ax-addrcl 9035  ax-mulcl 9036  ax-mulrcl 9037  ax-mulcom 9038  ax-addass 9039  ax-mulass 9040  ax-distr 9041  ax-i2m1 9042  ax-1ne0 9043  ax-1rid 9044  ax-rnegex 9045  ax-rrecex 9046  ax-cnre 9047  ax-pre-lttri 9048  ax-pre-lttrn 9049  ax-pre-ltadd 9050  ax-pre-mulgt0 9051  ax-pre-sup 9052  ax-hilex 22485  ax-hv0cl 22489  ax-hvaddid 22490  ax-hfvmul 22491  ax-hvmulid 22492  ax-hvmulass 22493  ax-hvmul0 22496  ax-hfi 22564  ax-his1 22567  ax-his3 22569  ax-his4 22570
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-reu 2699  df-rmo 2700  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-pss 3323  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-tp 3809  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-tr 4290  df-eprel 4481  df-id 4485  df-po 4490  df-so 4491  df-fr 4528  df-we 4530  df-ord 4571  df-on 4572  df-lim 4573  df-suc 4574  df-om 4832  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-2nd 6336  df-riota 6535  df-recs 6619  df-rdg 6654  df-er 6891  df-map 7006  df-en 7096  df-dom 7097  df-sdom 7098  df-sup 7432  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-sub 9277  df-neg 9278  df-div 9662  df-nn 9985  df-2 10042  df-3 10043  df-n0 10206  df-z 10267  df-uz 10473  df-rp 10597  df-seq 11307  df-exp 11366  df-cj 11887  df-re 11888  df-im 11889  df-sqr 12023  df-abs 12024  df-hnorm 22454  df-hvsub 22457  df-nmfn 23331  df-cnfn 23333  df-lnfn 23334
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