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Theorem nmcfnlbi 23543
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 5719 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmcfnex.1 . . . . . . 7  |-  T  e. 
LinFn
32lnfn0i 23533 . . . . . 6  |-  ( T `
 0h )  =  0
41, 3syl6eq 2483 . . . . 5  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
54abs00bd 12084 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
6 0le0 10070 . . . . 5  |-  0  <_  0
7 fveq2 5719 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
8 norm0 22618 . . . . . . . 8  |-  ( normh `  0h )  =  0
97, 8syl6eq 2483 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
109oveq2d 6088 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
11 nmcfnex.2 . . . . . . . . 9  |-  T  e. 
ConFn
122, 11nmcfnexi 23542 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1312recni 9091 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1413mul01i 9245 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1510, 14syl6req 2484 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
166, 15syl5breq 4239 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
175, 16eqbrtrd 4224 . . 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 )
) )
192lnfnfi 23532 . . . . . . . . . 10  |-  T : ~H
--> CC
2019ffvelrni 5860 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2120abscld 12226 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2221adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  RR )
2322recnd 9103 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  CC )
24 normcl 22615 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2524adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  RR )
2625recnd 9103 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  CC )
27 norm-i 22619 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =  0  <->  A  =  0h ) )
2827notbid 286 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( -.  ( normh `  A )  =  0  <->  -.  A  =  0h ) )
2928biimpar 472 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  -.  ( normh `  A )  =  0 )
3029neneqad 2668 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  =/=  0 )
3123, 26, 30divrec2d 9783 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) ) )
3225, 30rereccld 9830 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  RR )
3332recnd 9103 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  CC )
34 simpl 444 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  A  e.  ~H )
352lnfnmuli 23535 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3633, 34, 35syl2anc 643 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( T `  A
) ) )
3736fveq2d 5723 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
3820adantr 452 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  A )  e.  CC )
3933, 38absmuld 12244 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
40 df-ne 2600 . . . . . . . . . . . 12  |-  ( A  =/=  0h  <->  -.  A  =  0h )
41 normgt0 22617 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
4240, 41syl5bbr 251 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  <->  0  <  ( normh `  A
) ) )
4342biimpa 471 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( normh `  A ) )
4425, 43recgt0d 9934 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  (
1  /  ( normh `  A ) ) )
45 0re 9080 . . . . . . . . . 10  |-  0  e.  RR
46 ltle 9152 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
4745, 46mpan 652 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
4832, 44, 47sylc 58 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <_  (
1  /  ( normh `  A ) ) )
4932, 48absidd 12213 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
5049oveq1d 6087 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
5137, 39, 503eqtrrd 2472 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5231, 51eqtrd 2467 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
53 hvmulcl 22504 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5433, 34, 53syl2anc 643 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  .h  A
)  e.  ~H )
55 normcl 22615 . . . . . . 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 22739 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
5840, 57sylan2br 463 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
59 eqle 9165 . . . . . 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 643 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
61 nmfnlb 23415 . . . . . 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 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 ) )
6354, 60, 62syl2anc 643 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
6452, 63eqbrtrd 4224 . . 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 9876 . . . 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 1188 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( ( abs `  ( T `
 A ) )  /  ( normh `  A
) )  <_  ( normfn `
 T )  <->  ( abs `  ( T `  A
) )  <_  (
( normfn `  T )  x.  ( normh `  A )
) ) )
6864, 67mpbid 202 . 2  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
6918, 68pm2.61dan 767 1  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    x. cmul 8984    < clt 9109    <_ cle 9110    / cdiv 9666   abscabs 12027   ~Hchil 22410    .h csm 22412   normhcno 22414   0hc0v 22415   normfncnmf 22442   ConFnccnfn 22444   LinFnclf 22445
This theorem is referenced by:  nmcfnlb  23545
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-hilex 22490  ax-hv0cl 22494  ax-hvaddid 22495  ax-hfvmul 22496  ax-hvmulid 22497  ax-hvmulass 22498  ax-hvmul0 22501  ax-hfi 22569  ax-his1 22572  ax-his3 22574  ax-his4 22575
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-sup 7437  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-rp 10602  df-seq 11312  df-exp 11371  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-hnorm 22459  df-hvsub 22462  df-nmfn 23336  df-cnfn 23338  df-lnfn 23339
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