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Theorem nmcfnlbi 27640
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 5818 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmcfnex.1 . . . . . . 7  |-  T  e. 
LinFn
32lnfn0i 27630 . . . . . 6  |-  ( T `
 0h )  =  0
41, 3syl6eq 2472 . . . . 5  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
54abs00bd 13291 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
6 0le0 10643 . . . . 5  |-  0  <_  0
7 fveq2 5818 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
8 norm0 26716 . . . . . . . 8  |-  ( normh `  0h )  =  0
97, 8syl6eq 2472 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
109oveq2d 6258 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
11 nmcfnex.2 . . . . . . . . 9  |-  T  e. 
ConFn
122, 11nmcfnexi 27639 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1312recni 9599 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1413mul01i 9767 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1510, 14syl6req 2473 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
166, 15syl5breq 4395 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
175, 16eqbrtrd 4380 . . 3  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
1817adantl 467 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
192lnfnfi 27629 . . . . . . . . . 10  |-  T : ~H
--> CC
2019ffvelrni 5973 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2120abscld 13434 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2221adantr 466 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  RR )
2322recnd 9613 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  e.  CC )
24 normcl 26713 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2524adantr 466 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  RR )
2625recnd 9613 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  A
)  e.  CC )
27 norm-i 26717 . . . . . . . . 9  |-  ( A  e.  ~H  ->  (
( normh `  A )  =  0  <->  A  =  0h ) )
2827notbid 295 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( -.  ( normh `  A )  =  0  <->  -.  A  =  0h ) )
2928biimpar 487 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  -.  ( normh `  A )  =  0 )
3029neqned 2602 . . . . . 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 9613 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( 1  / 
( normh `  A )
)  e.  CC )
34 simpl 458 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  A  e.  ~H )
352lnfnmuli 27632 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3633, 34, 35syl2anc 665 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( T `  A
) ) )
3736fveq2d 5822 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
3820adantr 466 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( T `  A )  e.  CC )
3933, 38absmuld 13452 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
40 df-ne 2595 . . . . . . . . . . . 12  |-  ( A  =/=  0h  <->  -.  A  =  0h )
41 normgt0 26715 . . . . . . . . . . . 12  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
4240, 41syl5bbr 262 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( -.  A  =  0h  <->  0  <  ( normh `  A
) ) )
4342biimpa 486 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  ( normh `  A ) )
4425, 43recgt0d 10485 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <  (
1  /  ( normh `  A ) ) )
45 0re 9587 . . . . . . . . . 10  |-  0  e.  RR
46 ltle 9666 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
4745, 46mpan 674 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
4832, 44, 47sylc 62 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  0  <_  (
1  /  ( normh `  A ) ) )
4932, 48absidd 13421 . . . . . . 7  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
5049oveq1d 6257 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
5137, 39, 503eqtrrd 2461 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  x.  ( abs `  ( T `  A ) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5231, 51eqtrd 2456 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
53 hvmulcl 26601 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5433, 34, 53syl2anc 665 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( 1  /  ( normh `  A
) )  .h  A
)  e.  ~H )
55 normcl 26713 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5654, 55syl 17 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
57 norm1 26837 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
5840, 57sylan2br 478 . . . . . 6  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  1 )
59 eqle 9680 . . . . . 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 665 . . . . 5  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
61 nmfnlb 27512 . . . . . 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 1347 . . . . 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 665 . . . 4  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
6452, 63eqbrtrd 4380 . . 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 10428 . . . 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 1268 . . 3  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( ( ( abs `  ( T `
 A ) )  /  ( normh `  A
) )  <_  ( normfn `
 T )  <->  ( abs `  ( T `  A
) )  <_  (
( normfn `  T )  x.  ( normh `  A )
) ) )
6864, 67mpbid 213 . 2  |-  ( ( A  e.  ~H  /\  -.  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
6918, 68pm2.61dan 798 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2593   class class class wbr 4359   -->wf 5533   ` cfv 5537  (class class class)co 6242   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    x. cmul 9488    < clt 9619    <_ cle 9620    / cdiv 10213   abscabs 13234   ~Hchil 26507    .h csm 26509   normhcno 26511   0hc0v 26512   normfncnmf 26539   ConFnccnfn 26541   LinFnclf 26542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-hilex 26587  ax-hv0cl 26591  ax-hvaddid 26592  ax-hfvmul 26593  ax-hvmulid 26594  ax-hvmulass 26595  ax-hvmul0 26598  ax-hfi 26667  ax-his1 26670  ax-his3 26672  ax-his4 26673
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-om 6644  df-2nd 6745  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-er 7311  df-map 7422  df-en 7518  df-dom 7519  df-sdom 7520  df-sup 7902  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-div 10214  df-nn 10554  df-2 10612  df-3 10613  df-n0 10814  df-z 10882  df-uz 11104  df-rp 11247  df-seq 12157  df-exp 12216  df-cj 13099  df-re 13100  df-im 13101  df-sqrt 13235  df-abs 13236  df-hnorm 26556  df-hvsub 26559  df-nmfn 27433  df-cnfn 27435  df-lnfn 27436
This theorem is referenced by:  nmcfnlb  27642
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