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Theorem nmbdfnlb 25626
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 5801 . . . . . 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 5806 . . . . 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 5802 . . . . . 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 6218 . . . . 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 4416 . . . 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 316 . . 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 2526 . . . . . 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 2523 . . . . . 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 710 . . . . 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 2526 . . . . . 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 5802 . . . . . . 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 2523 . . . . . 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 710 . . . . 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 25561 . . . . . 6  |-  ( ~H 
X.  { 0 } )  e.  LinFn
15 nmfn0 25563 . . . . . . 7  |-  ( normfn `  ( ~H  X.  {
0 } ) )  =  0
16 0re 9500 . . . . . . 7  |-  0  e.  RR
1715, 16eqeltri 2538 . . . . . 6  |-  ( normfn `  ( ~H  X.  {
0 } ) )  e.  RR
1814, 17pm3.2i 455 . . . . 5  |-  ( ( ~H  X.  { 0 } )  e.  LinFn  /\  ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR )
199, 13, 18elimhyp 3959 . . . 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 25625 . . 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 3952 . 2  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  ->  ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) ) )
22213impia 1185 1  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR  /\  A  e. 
~H )  ->  ( abs `  ( T `  A ) )  <_ 
( ( normfn `  T
)  x.  ( normh `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ifcif 3902   {csn 3988   class class class wbr 4403    X. cxp 4949   ` cfv 5529  (class class class)co 6203   RRcr 9395   0cc0 9396    x. cmul 9401    <_ cle 9533   abscabs 12844   ~Hchil 24493   normhcno 24497   normfncnmf 24525   LinFnclf 24528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-hilex 24573  ax-hfvadd 24574  ax-hv0cl 24577  ax-hvaddid 24578  ax-hfvmul 24579  ax-hvmulid 24580  ax-hvmul0 24584  ax-hfi 24653  ax-his1 24656  ax-his3 24658  ax-his4 24659
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-hnorm 24542  df-nmfn 25421  df-lnfn 25424
This theorem is referenced by:  lnfncnbd  25633
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