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Theorem nmbdfnlbi 22942
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 5632 . . . . . . 7  |-  ( A  =  0h  ->  ( T `  A )  =  ( T `  0h ) )
2 nmbdfnlb.1 . . . . . . . . 9  |-  ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR )
32simpli 444 . . . . . . . 8  |-  T  e. 
LinFn
43lnfn0i 22935 . . . . . . 7  |-  ( T `
 0h )  =  0
51, 4syl6eq 2414 . . . . . 6  |-  ( A  =  0h  ->  ( T `  A )  =  0 )
65fveq2d 5636 . . . . 5  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  ( abs `  0
) )
7 abs0 11977 . . . . 5  |-  ( abs `  0 )  =  0
86, 7syl6eq 2414 . . . 4  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  =  0 )
9 0le0 9974 . . . . 5  |-  0  <_  0
10 fveq2 5632 . . . . . . . 8  |-  ( A  =  0h  ->  ( normh `  A )  =  ( normh `  0h )
)
11 norm0 22020 . . . . . . . 8  |-  ( normh `  0h )  =  0
1210, 11syl6eq 2414 . . . . . . 7  |-  ( A  =  0h  ->  ( normh `  A )  =  0 )
1312oveq2d 5997 . . . . . 6  |-  ( A  =  0h  ->  (
( normfn `  T )  x.  ( normh `  A )
)  =  ( (
normfn `  T )  x.  0 ) )
142simpri 448 . . . . . . . 8  |-  ( normfn `  T )  e.  RR
1514recni 8996 . . . . . . 7  |-  ( normfn `  T )  e.  CC
1615mul01i 9149 . . . . . 6  |-  ( (
normfn `  T )  x.  0 )  =  0
1713, 16syl6req 2415 . . . . 5  |-  ( A  =  0h  ->  0  =  ( ( normfn `  T )  x.  ( normh `  A ) ) )
189, 17syl5breq 4160 . . . 4  |-  ( A  =  0h  ->  0  <_  ( ( normfn `  T
)  x.  ( normh `  A ) ) )
198, 18eqbrtrd 4145 . . 3  |-  ( A  =  0h  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
2019adantl 452 . 2  |-  ( ( A  e.  ~H  /\  A  =  0h )  ->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
213lnfnfi 22934 . . . . . . . . . 10  |-  T : ~H
--> CC
2221ffvelrni 5771 . . . . . . . . 9  |-  ( A  e.  ~H  ->  ( T `  A )  e.  CC )
2322abscld 12125 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  e.  RR )
2423adantr 451 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  e.  RR )
2524recnd 9008 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  e.  CC )
26 normcl 22017 . . . . . . . 8  |-  ( A  e.  ~H  ->  ( normh `  A )  e.  RR )
2726adantr 451 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  RR )
2827recnd 9008 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  e.  CC )
29 normne0 22022 . . . . . . 7  |-  ( A  e.  ~H  ->  (
( normh `  A )  =/=  0  <->  A  =/=  0h )
)
3029biimpar 471 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  A )  =/=  0 )
3125, 28, 30divrec2d 9687 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) ) )
3227, 30rereccld 9734 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  RR )
3332recnd 9008 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( 1  /  ( normh `  A ) )  e.  CC )
34 simpl 443 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  ->  A  e.  ~H )
353lnfnmuli 22937 . . . . . . . 8  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  / 
( normh `  A )
)  x.  ( T `
 A ) ) )
3633, 34, 35syl2anc 642 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  (
( 1  /  ( normh `  A ) )  .h  A ) )  =  ( ( 1  /  ( normh `  A
) )  x.  ( T `  A )
) )
3736fveq2d 5636 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  =  ( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) ) )
3822adantr 451 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( T `  A
)  e.  CC )
3933, 38absmuld 12143 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
( 1  /  ( normh `  A ) )  x.  ( T `  A ) ) )  =  ( ( abs `  ( 1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A
) ) ) )
40 normgt0 22019 . . . . . . . . . . 11  |-  ( A  e.  ~H  ->  ( A  =/=  0h  <->  0  <  (
normh `  A ) ) )
4140biimpa 470 . . . . . . . . . 10  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( normh `  A ) )
4227, 41recgt0d 9838 . . . . . . . . 9  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <  ( 1  /  ( normh `  A
) ) )
43 0re 8985 . . . . . . . . . 10  |-  0  e.  RR
44 ltle 9057 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  ( 1  /  ( normh `  A ) )  e.  RR )  -> 
( 0  <  (
1  /  ( normh `  A ) )  -> 
0  <_  ( 1  /  ( normh `  A
) ) ) )
4543, 44mpan 651 . . . . . . . . 9  |-  ( ( 1  /  ( normh `  A ) )  e.  RR  ->  ( 0  <  ( 1  / 
( normh `  A )
)  ->  0  <_  ( 1  /  ( normh `  A ) ) ) )
4632, 42, 45sylc 56 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
0  <_  ( 1  /  ( normh `  A
) ) )
4732, 46absidd 12112 . . . . . . 7  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  (
1  /  ( normh `  A ) ) )  =  ( 1  / 
( normh `  A )
) )
4847oveq1d 5996 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  (
1  /  ( normh `  A ) ) )  x.  ( abs `  ( T `  A )
) )  =  ( ( 1  /  ( normh `  A ) )  x.  ( abs `  ( T `  A )
) ) )
4937, 39, 483eqtrrd 2403 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  x.  ( abs `  ( T `  A
) ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
5031, 49eqtrd 2398 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  =  ( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) ) )
51 hvmulcl 21906 . . . . . 6  |-  ( ( ( 1  /  ( normh `  A ) )  e.  CC  /\  A  e.  ~H )  ->  (
( 1  /  ( normh `  A ) )  .h  A )  e. 
~H )
5233, 34, 51syl2anc 642 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H )
53 normcl 22017 . . . . . . 7  |-  ( ( ( 1  /  ( normh `  A ) )  .h  A )  e. 
~H  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
5452, 53syl 15 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR )
55 norm1 22141 . . . . . 6  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  =  1 )
56 eqle 9070 . . . . . 6  |-  ( ( ( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  e.  RR  /\  ( normh `  ( ( 1  / 
( normh `  A )
)  .h  A ) )  =  1 )  ->  ( normh `  (
( 1  /  ( normh `  A ) )  .h  A ) )  <_  1 )
5754, 55, 56syl2anc 642 . . . . 5  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normh `  ( (
1  /  ( normh `  A ) )  .h  A ) )  <_ 
1 )
58 nmfnlb 22817 . . . . . 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 ) )
5921, 58mp3an1 1265 . . . . 5  |-  ( ( ( ( 1  / 
( normh `  A )
)  .h  A )  e.  ~H  /\  ( normh `  ( ( 1  /  ( normh `  A
) )  .h  A
) )  <_  1
)  ->  ( abs `  ( T `  (
( 1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T ) )
6052, 57, 59syl2anc 642 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  ( (
1  /  ( normh `  A ) )  .h  A ) ) )  <_  ( normfn `  T
) )
6150, 60eqbrtrd 4145 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( abs `  ( T `  A )
)  /  ( normh `  A ) )  <_ 
( normfn `  T )
)
6214a1i 10 . . . 4  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( normfn `  T )  e.  RR )
63 ledivmul2 9780 . . . 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 ) ) ) )
6424, 62, 27, 41, 63syl112anc 1187 . . 3  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( ( ( abs `  ( T `  A
) )  /  ( normh `  A ) )  <_  ( normfn `  T
)  <->  ( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) ) )
6561, 64mpbid 201 . 2  |-  ( ( A  e.  ~H  /\  A  =/=  0h )  -> 
( abs `  ( T `  A )
)  <_  ( ( normfn `
 T )  x.  ( normh `  A )
) )
6620, 65pm2.61dane 2607 1  |-  ( A  e.  ~H  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   -->wf 5354   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885    x. cmul 8889    < clt 9014    <_ cle 9015    / cdiv 9570   abscabs 11926   ~Hchil 21812    .h csm 21814   normhcno 21816   0hc0v 21817   normfncnmf 21844   LinFnclf 21847
This theorem is referenced by:  nmbdfnlb  22943
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-hilex 21892  ax-hv0cl 21896  ax-hvaddid 21897  ax-hfvmul 21898  ax-hvmulid 21899  ax-hvmul0 21903  ax-hfi 21971  ax-his1 21974  ax-his3 21976  ax-his4 21977
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-seq 11211  df-exp 11270  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-hnorm 21861  df-nmfn 22738  df-lnfn 22741
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