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Theorem nmbdfnlb 23510
Description: A lower bound for the norm of a bounded linear functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)
Assertion
Ref Expression
nmbdfnlb  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR  /\  A  e. 
~H )  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )

Proof of Theorem nmbdfnlb
StepHypRef Expression
1 fveq1 5690 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( T `  A )  =  ( if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) ) `
 A ) )
21fveq2d 5695 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( abs `  ( T `  A
) )  =  ( abs `  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) ) )
3 fveq2 5691 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( normfn `  T )  =  (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) ) )
43oveq1d 6059 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( normfn `
 T )  x.  ( normh `  A )
)  =  ( (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) )
52, 4breq12d 4189 . . . 4  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) )  <->  ( abs `  ( if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_  ( ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) ) )
65imbi2d 308 . . 3  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) )  <->  ( A  e. 
~H  ->  ( abs `  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_ 
( ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A
) ) ) ) )
7 eleq1 2468 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( T  e.  LinFn 
<->  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) )  e.  LinFn ) )
83eleq1d 2474 . . . . . 6  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( normfn `
 T )  e.  RR  <->  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) )
97, 8anbi12d 692 . . . . 5  |-  ( T  =  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  ->  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  <->  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e. 
LinFn  /\  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) ) )
10 eleq1 2468 . . . . . 6  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( ~H  X.  { 0 } )  e.  LinFn 
<->  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) )  e.  LinFn ) )
11 fveq2 5691 . . . . . . 7  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( normfn `  ( ~H  X.  { 0 } ) )  =  ( normfn `  if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T , 
( ~H  X.  {
0 } ) ) ) )
1211eleq1d 2474 . . . . . 6  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR  <->  (
normfn `  if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) )
1310, 12anbi12d 692 . . . . 5  |-  ( ( ~H  X.  { 0 } )  =  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  -> 
( ( ( ~H 
X.  { 0 } )  e.  LinFn  /\  ( normfn `
 ( ~H  X.  { 0 } ) )  e.  RR )  <-> 
( if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e.  LinFn  /\  ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR ) ) )
14 0lnfn 23445 . . . . . 6  |-  ( ~H 
X.  { 0 } )  e.  LinFn
15 nmfn0 23447 . . . . . . 7  |-  ( normfn `  ( ~H  X.  {
0 } ) )  =  0
16 0re 9051 . . . . . . 7  |-  0  e.  RR
1715, 16eqeltri 2478 . . . . . 6  |-  ( normfn `  ( ~H  X.  {
0 } ) )  e.  RR
1814, 17pm3.2i 442 . . . . 5  |-  ( ( ~H  X.  { 0 } )  e.  LinFn  /\  ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR )
199, 13, 18elimhyp 3751 . . . 4  |-  ( if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) )  e. 
LinFn  /\  ( normfn `  if ( ( T  e. 
LinFn  /\  ( normfn `  T
)  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  e.  RR )
2019nmbdfnlbi 23509 . . 3  |-  ( A  e.  ~H  ->  ( abs `  ( if ( ( T  e.  LinFn  /\  ( normfn `  T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) `  A ) )  <_  ( ( normfn `
 if ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR ) ,  T ,  ( ~H  X.  { 0 } ) ) )  x.  ( normh `  A ) ) )
216, 20dedth 3744 . 2  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  ->  ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) ) )
22213impia 1150 1  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR  /\  A  e. 
~H )  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   ifcif 3703   {csn 3778   class class class wbr 4176    X. cxp 4839   ` cfv 5417  (class class class)co 6044   RRcr 8949   0cc0 8950    x. cmul 8955    <_ cle 9081   abscabs 11998   ~Hchil 22379   normhcno 22383   normfncnmf 22411   LinFnclf 22414
This theorem is referenced by:  lnfncnbd  23517
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-hilex 22459  ax-hfvadd 22460  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvmul0 22470  ax-hfi 22538  ax-his1 22541  ax-his3 22543  ax-his4 22544
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-seq 11283  df-exp 11342  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-hnorm 22428  df-nmfn 23305  df-lnfn 23308
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