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Theorem nmbdfnlbi 23552
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
nmbdfnlb.1  |-  ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR )
Assertion
Ref Expression
nmbdfnlbi  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmbdfnlbi
StepHypRef Expression
1 fveq2 5728 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmbdfnlb.1 . . . . . . . 8  |-  ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR )
32simpli 445 . . . . . . 7  |-  T  e. 
LinFn
43lnfn0i 23545 . . . . . 6  |-  ( T `
 0h )  =  0
51, 4syl6eq 2484 . . . . 5  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
65abs00bd 12096 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
7 0le0 10081 . . . . 5  |-  0  <_  0
8 fveq2 5728 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
9 norm0 22630 . . . . . . . 8  |-  ( normh `  0h )  =  0
108, 9syl6eq 2484 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1110oveq2d 6097 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
122simpri 449 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1312recni 9102 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1413mul01i 9256 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1511, 14syl6req 2485 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
167, 15syl5breq 4247 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
176, 16eqbrtrd 4232 . . 3  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
1817adantl 453 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
193lnfnfi 23544 . . . . . . . . . 10  |-  T : ~H
--> CC
2019ffvelrni 5869 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2120abscld 12238 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2221adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  e.  RR )
2322recnd 9114 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  e.  CC )
24 normcl 22627 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2524adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
2625recnd 9114 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
27 normne0 22632 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
2827biimpar 472 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
2923, 26, 28divrec2d 9794 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
3025, 28rereccld 9841 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
3130recnd 9114 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
32 simpl 444 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
333lnfnmuli 23547 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3431, 32, 33syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( T `  A )
) )
3534fveq2d 5732 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
3620adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  CC )
3731, 36absmuld 12256 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
38 normgt0 22629 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
3938biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
4025, 39recgt0d 9945 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
41 0re 9091 . . . . . . . . . 10  |-  0  e.  RR
42 ltle 9163 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
4341, 42mpan 652 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
4430, 40, 43sylc 58 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
4530, 44absidd 12225 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
4645oveq1d 6096 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( abs `  ( T `  A )
) ) )
4735, 37, 463eqtrrd 2473 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
4829, 47eqtrd 2468 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
49 hvmulcl 22516 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5031, 32, 49syl2anc 643 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
51 normcl 22627 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5250, 51syl 16 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
53 norm1 22751 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
54 eqle 9176 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
5552, 53, 54syl2anc 643 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
56 nmfnlb 23427 . . . . . 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 ) )
5719, 56mp3an1 1266 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( abs `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T ) )
5850, 55, 57syl2anc 643 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
5948, 58eqbrtrd 4232 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normfn `  T )
)
6012a1i 11 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normfn `  T )  e.  RR )
61 ledivmul2 9887 . . . 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 ) ) ) )
6222, 60, 25, 39, 61syl112anc 1188 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normfn `  T
)  <->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) ) )
6359, 62mpbid 202 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
6418, 63pm2.61dane 2682 1  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990   1c1 8991    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677   abscabs 12039   ~Hchil 22422    .h csm 22424   normhcno 22426   0hc0v 22427   normfncnmf 22454   LinFnclf 22457
This theorem is referenced by:  nmbdfnlb  23553
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-hilex 22502  ax-hv0cl 22506  ax-hvaddid 22507  ax-hfvmul 22508  ax-hvmulid 22509  ax-hvmul0 22513  ax-hfi 22581  ax-his1 22584  ax-his3 22586  ax-his4 22587
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-hnorm 22471  df-nmfn 23348  df-lnfn 23351
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