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

Theorem nmcfnlbi 26647
Description: A lower bound for the norm of a continuous linear functional. Theorem 3.5(ii) of [Beran] p. 99. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nmcfnex.1  |-  T  e. 
LinFn
nmcfnex.2  |-  T  e. 
ConFn
Assertion
Ref Expression
nmcfnlbi  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmcfnlbi
StepHypRef Expression
1 fveq2 5864 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmcfnex.1 . . . . . . 7  |-  T  e. 
LinFn
32lnfn0i 26637 . . . . . 6  |-  ( T `
 0h )  =  0
41, 3syl6eq 2524 . . . . 5  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
54abs00bd 13083 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
6 0le0 10621 . . . . 5  |-  0  <_  0
7 fveq2 5864 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
8 norm0 25721 . . . . . . . 8  |-  ( normh `  0h )  =  0
97, 8syl6eq 2524 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
109oveq2d 6298 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
11 nmcfnex.2 . . . . . . . . 9  |-  T  e. 
ConFn
122, 11nmcfnexi 26646 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1312recni 9604 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1413mul01i 9765 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1510, 14syl6req 2525 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
166, 15syl5breq 4482 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
175, 16eqbrtrd 4467 . . 3  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
1817adantl 466 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
192lnfnfi 26636 . . . . . . . . . 10  |-  T : ~H
--> CC
2019ffvelrni 6018 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2120abscld 13226 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2221adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  RR )
2322recnd 9618 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  CC )
24 normcl 25718 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2524adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  RR )
2625recnd 9618 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  CC )
27 norm-i 25722 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =  0  <->  A  =  0h ) )
2827notbid 294 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( -.  ( normh `  A )  =  0  <->  -.  A  =  0h ) )
2928biimpar 485 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  -.  ( normh `  A )  =  0 )
3029neqned 2670 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  =/=  0 )
3123, 26, 30divrec2d 10320 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) ) )
3225, 30rereccld 10367 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  RR )
3332recnd 9618 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  CC )
34 simpl 457 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  A  e.  ~H )
352lnfnmuli 26639 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3633, 34, 35syl2anc 661 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( T `  A
) ) )
3736fveq2d 5868 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
3820adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  A )  e.  CC )
3933, 38absmuld 13244 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
40 df-ne 2664 . . . . . . . . . . . 12  |-  ( A  =/=  0h  <->  -.  A  =  0h )
41 normgt0 25720 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
4240, 41syl5bbr 259 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  <->  0  <  ( normh `  A
) ) )
4342biimpa 484 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( normh `  A ) )
4425, 43recgt0d 10476 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  (
1  /  ( normh `  A ) ) )
45 0re 9592 . . . . . . . . . 10  |-  0  e.  RR
46 ltle 9669 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
4745, 46mpan 670 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
4832, 44, 47sylc 60 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <_  (
1  /  ( normh `  A ) ) )
4932, 48absidd 13213 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
5049oveq1d 6297 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
5137, 39, 503eqtrrd 2513 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5231, 51eqtrd 2508 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
53 hvmulcl 25606 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5433, 34, 53syl2anc 661 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  .h  A
)  e.  ~H )
55 normcl 25718 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5654, 55syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
57 norm1 25843 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
5840, 57sylan2br 476 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
59 eqle 9683 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
6056, 58, 59syl2anc 661 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
61 nmfnlb 26519 . . . . . 6  |-  ( ( T : ~H --> CC  /\  ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( abs `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T ) )
6219, 61mp3an1 1311 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( abs `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T ) )
6354, 60, 62syl2anc 661 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
6452, 63eqbrtrd 4467 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normfn `  T
) )
6512a1i 11 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normfn `  T
)  e.  RR )
66 ledivmul2 10418 . . . 4  |-  ( ( ( abs `  ( T `  A )
)  e.  RR  /\  ( normfn `  T )  e.  RR  /\  ( (
normh `  A )  e.  RR  /\  0  < 
( normh `  A )
) )  ->  (
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normfn `  T )  <->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) ) )
6722, 65, 25, 43, 66syl112anc 1232 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( ( abs `  ( T `
 A ) )  /  ( normh `  A
) )  <_  ( normfn `
 T )  <->  ( abs `  ( T `  A
) )  <_  (
( normfn `  T )  x.  ( normh `  A )
) ) )
6864, 67mpbid 210 . 2  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
6918, 68pm2.61dan 789 1  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    <_ cle 9625    / cdiv 10202   abscabs 13026   ~Hchil 25512    .h csm 25514   normhcno 25516   0hc0v 25517   normfncnmf 25544   ConFnccnfn 25546   LinFnclf 25547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-hilex 25592  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his3 25677  ax-his4 25678
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12072  df-exp 12131  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-hnorm 25561  df-hvsub 25564  df-nmfn 26440  df-cnfn 26442  df-lnfn 26443
This theorem is referenced by:  nmcfnlb  26649
  Copyright terms: Public domain W3C validator