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

Theorem nmcfnlbi 22462
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 5377 . . . . . . 7  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmcfnex.1 . . . . . . . 8  |-  T  e. 
LinFn
32lnfn0i 22452 . . . . . . 7  |-  ( T `
 0h )  =  0
41, 3syl6eq 2301 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
54fveq2d 5381 . . . . 5  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  ( abs `  0
) )
6 abs0 11647 . . . . 5  |-  ( abs `  0 )  =  0
75, 6syl6eq 2301 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
8 0le0 9707 . . . . 5  |-  0  <_  0
9 fveq2 5377 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
10 norm0 21537 . . . . . . . 8  |-  ( normh `  0h )  =  0
119, 10syl6eq 2301 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1211oveq2d 5726 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
13 nmcfnex.2 . . . . . . . . 9  |-  T  e. 
ConFn
142, 13nmcfnexi 22461 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1514recni 8729 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1615mul01i 8882 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1712, 16syl6req 2302 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
188, 17syl5breq 3955 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
197, 18eqbrtrd 3940 . . 3  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
2019adantl 454 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
212lnfnfi 22451 . . . . . . . . . 10  |-  T : ~H
--> CC
2221ffvelrni 5516 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2322abscld 11795 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2423adantr 453 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  RR )
2524recnd 8741 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  CC )
26 normcl 21534 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2726adantr 453 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  RR )
2827recnd 8741 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  CC )
29 norm-i 21538 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =  0  <->  A  =  0h ) )
3029notbid 287 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( -.  ( normh `  A )  =  0  <->  -.  A  =  0h ) )
3130biimpar 473 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  -.  ( normh `  A )  =  0 )
32 df-ne 2414 . . . . . . 7  |-  ( (
normh `  A )  =/=  0  <->  -.  ( normh `  A )  =  0 )
3331, 32sylibr 205 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  =/=  0 )
3425, 28, 33divrec2d 9420 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) ) )
3527, 33rereccld 9467 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  RR )
3635recnd 8741 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  CC )
37 simpl 445 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  A  e.  ~H )
382lnfnmuli 22454 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3936, 37, 38syl2anc 645 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( T `  A
) ) )
4039fveq2d 5381 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
4122adantr 453 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  A )  e.  CC )
4236, 41absmuld 11813 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
43 df-ne 2414 . . . . . . . . . . . 12  |-  ( A  =/=  0h  <->  -.  A  =  0h )
44 normgt0 21536 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
4543, 44syl5bbr 252 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  <->  0  <  ( normh `  A
) ) )
4645biimpa 472 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( normh `  A ) )
4727, 46recgt0d 9571 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  (
1  /  ( normh `  A ) ) )
48 0re 8718 . . . . . . . . . 10  |-  0  e.  RR
49 ltle 8790 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
5048, 49mpan 654 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
5135, 47, 50sylc 58 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <_  (
1  /  ( normh `  A ) ) )
5235, 51absidd 11782 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
5352oveq1d 5725 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
5440, 42, 533eqtrrd 2290 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5534, 54eqtrd 2285 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
56 hvmulcl 21423 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5736, 37, 56syl2anc 645 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  .h  A
)  e.  ~H )
58 normcl 21534 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5957, 58syl 17 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
60 norm1 21658 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
6143, 60sylan2br 464 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
62 eqle 8803 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
6359, 61, 62syl2anc 645 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
64 nmfnlb 22334 . . . . . 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 ) )
6521, 64mp3an1 1269 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( abs `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T ) )
6657, 63, 65syl2anc 645 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
6755, 66eqbrtrd 3940 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normfn `  T
) )
6814a1i 12 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normfn `  T
)  e.  RR )
69 ledivmul2 9513 . . . 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 ) ) ) )
7024, 68, 27, 46, 69syl112anc 1191 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( ( abs `  ( T `
 A ) )  /  ( normh `  A
) )  <_  ( normfn `
 T )  <->  ( abs `  ( T `  A
) )  <_  (
( normfn `  T )  x.  ( normh `  A )
) ) )
7167, 70mpbid 203 . 2  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
7220, 71pm2.61dan 769 1  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   -->wf 4588   ` cfv 4592  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    x. cmul 8622    < clt 8747    <_ cle 8748    / cdiv 9303   abscabs 11596   ~Hchil 21329    .h csm 21331   normhcno 21333   0hc0v 21334   normfncnmf 21361   ConFnccnfn 21363   LinFnclf 21364
This theorem is referenced by:  nmcfnlb  22464
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-hilex 21409  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-hnorm 21378  df-hvsub 21381  df-nmfn 22255  df-cnfn 22257  df-lnfn 22258
  Copyright terms: Public domain W3C validator