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Theorem nmcfnlbi 26949
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 5856 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmcfnex.1 . . . . . . 7  |-  T  e. 
LinFn
32lnfn0i 26939 . . . . . 6  |-  ( T `
 0h )  =  0
41, 3syl6eq 2500 . . . . 5  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
54abs00bd 13106 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
6 0le0 10632 . . . . 5  |-  0  <_  0
7 fveq2 5856 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
8 norm0 26023 . . . . . . . 8  |-  ( normh `  0h )  =  0
97, 8syl6eq 2500 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
109oveq2d 6297 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
11 nmcfnex.2 . . . . . . . . 9  |-  T  e. 
ConFn
122, 11nmcfnexi 26948 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1312recni 9611 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1413mul01i 9773 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1510, 14syl6req 2501 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
166, 15syl5breq 4472 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
175, 16eqbrtrd 4457 . . 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 26938 . . . . . . . . . 10  |-  T : ~H
--> CC
2019ffvelrni 6015 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2120abscld 13249 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2221adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  RR )
2322recnd 9625 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  CC )
24 normcl 26020 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2524adantr 465 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  RR )
2625recnd 9625 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  CC )
27 norm-i 26024 . . . . . . . . 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 2646 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  =/=  0 )
3123, 26, 30divrec2d 10331 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) ) )
3225, 30rereccld 10378 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  RR )
3332recnd 9625 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  CC )
34 simpl 457 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  A  e.  ~H )
352lnfnmuli 26941 . . . . . . . 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 5860 . . . . . 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 13267 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
40 df-ne 2640 . . . . . . . . . . . 12  |-  ( A  =/=  0h  <->  -.  A  =  0h )
41 normgt0 26022 . . . . . . . . . . . 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 10487 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  (
1  /  ( normh `  A ) ) )
45 0re 9599 . . . . . . . . . 10  |-  0  e.  RR
46 ltle 9676 . . . . . . . . . 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 13236 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
5049oveq1d 6296 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
5137, 39, 503eqtrrd 2489 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5231, 51eqtrd 2484 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
53 hvmulcl 25908 . . . . . 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 26020 . . . . . . 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 26145 . . . . . . 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 9690 . . . . . 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 26821 . . . . . 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 1312 . . . . 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 4457 . . 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 10429 . . . 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 1233 . . 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 791 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 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    x. cmul 9500    < clt 9631    <_ cle 9632    / cdiv 10213   abscabs 13049   ~Hchil 25814    .h csm 25816   normhcno 25818   0hc0v 25819   normfncnmf 25846   ConFnccnfn 25848   LinFnclf 25849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-hilex 25894  ax-hv0cl 25898  ax-hvaddid 25899  ax-hfvmul 25900  ax-hvmulid 25901  ax-hvmulass 25902  ax-hvmul0 25905  ax-hfi 25974  ax-his1 25977  ax-his3 25979  ax-his4 25980
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11093  df-rp 11232  df-seq 12090  df-exp 12149  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-hnorm 25863  df-hvsub 25866  df-nmfn 26742  df-cnfn 26744  df-lnfn 26745
This theorem is referenced by:  nmcfnlb  26951
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