HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nmbdfnlb Unicode version

Theorem nmbdfnlb 22744
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmbdfnlb  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR  /\  A  e. 
~H )  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmbdfnlb
StepHypRef Expression
1 fveq1 5607 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( T `  A )  =  ( if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) ) `
 A ) )
21fveq2d 5612 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( abs `  ( T `  A
) )  =  ( abs `  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) ) )
3 fveq2 5608 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( normfn `  T )  =  (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) ) )
43oveq1d 5960 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( normfn `
 T )  x.  ( normh `  A )
)  =  ( (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) )
52, 4breq12d 4117 . . . 4  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) )  <->  ( abs `  ( if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_  ( ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) ) )
65imbi2d 307 . . 3  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) )  <->  ( A  e. 
~H  ->  ( abs `  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_ 
( ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A
) ) ) ) )
7 eleq1 2418 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( T  e.  LinFn 
<->  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) )  e.  LinFn ) )
83eleq1d 2424 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( normfn `
 T )  e.  RR  <->  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) )
97, 8anbi12d 691 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  <->  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e. 
LinFn  /\  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) ) )
10 eleq1 2418 . . . . . 6  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( ~H  X.  { 0 } )  e.  LinFn 
<->  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) )  e.  LinFn ) )
11 fveq2 5608 . . . . . . 7  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( normfn `  ( ~H  X.  { 0 } ) )  =  ( normfn `  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) ) ) )
1211eleq1d 2424 . . . . . 6  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR  <->  (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) )
1310, 12anbi12d 691 . . . . 5  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( ( ~H 
X.  { 0 } )  e.  LinFn  /\  ( normfn `
 ( ~H  X.  { 0 } ) )  e.  RR )  <-> 
( if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e.  LinFn  /\  ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) ) )
14 0lnfn 22679 . . . . . 6  |-  ( ~H 
X.  { 0 } )  e.  LinFn
15 nmfn0 22681 . . . . . . 7  |-  ( normfn `  ( ~H  X.  {
0 } ) )  =  0
16 0re 8928 . . . . . . 7  |-  0  e.  RR
1715, 16eqeltri 2428 . . . . . 6  |-  ( normfn `  ( ~H  X.  {
0 } ) )  e.  RR
1814, 17pm3.2i 441 . . . . 5  |-  ( ( ~H  X.  { 0 } )  e.  LinFn  /\  ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR )
199, 13, 18elimhyp 3689 . . . 4  |-  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e. 
LinFn  /\  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR )
2019nmbdfnlbi 22743 . . 3  |-  ( A  e.  ~H  ->  ( abs `  ( if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_  ( ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) )
216, 20dedth 3682 . 2  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  ->  ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) ) )
22213impia 1148 1  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR  /\  A  e. 
~H )  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   ifcif 3641   {csn 3716   class class class wbr 4104    X. cxp 4769   ` cfv 5337  (class class class)co 5945   RRcr 8826   0cc0 8827    x. cmul 8832    <_ cle 8958   abscabs 11815   ~Hchil 21613   normhcno 21617   normfncnmf 21645   LinFnclf 21648
This theorem is referenced by:  lnfncnbd  22751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-hilex 21693  ax-hfvadd 21694  ax-hv0cl 21697  ax-hvaddid 21698  ax-hfvmul 21699  ax-hvmulid 21700  ax-hvmul0 21704  ax-hfi 21772  ax-his1 21775  ax-his3 21777  ax-his4 21778
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-seq 11139  df-exp 11198  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-hnorm 21662  df-nmfn 22539  df-lnfn 22542
  Copyright terms: Public domain W3C validator