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Theorem nmbdfnlb 27263
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 5802 . . . . . 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 5807 . . . . 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 5803 . . . . . 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 6247 . . . . 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 4405 . . . 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 314 . . 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 2472 . . . . . 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 2469 . . . . . 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 709 . . . . 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 2472 . . . . . 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 5803 . . . . . . 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 2469 . . . . . 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 709 . . . . 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 27198 . . . . . 6  |-  ( ~H 
X.  { 0 } )  e.  LinFn
15 nmfn0 27200 . . . . . . 7  |-  ( normfn `  ( ~H  X.  {
0 } ) )  =  0
16 0re 9544 . . . . . . 7  |-  0  e.  RR
1715, 16eqeltri 2484 . . . . . 6  |-  ( normfn `  ( ~H  X.  {
0 } ) )  e.  RR
1814, 17pm3.2i 453 . . . . 5  |-  ( ( ~H  X.  { 0 } )  e.  LinFn  /\  ( normfn `  ( ~H  X.  { 0 } ) )  e.  RR )
199, 13, 18elimhyp 3940 . . . 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 27262 . . 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 3933 . 2  |-  ( ( T  e.  LinFn  /\  ( normfn `
 T )  e.  RR )  ->  ( A  e.  ~H  ->  ( abs `  ( T `
 A ) )  <_  ( ( normfn `  T )  x.  ( normh `  A ) ) ) )
22213impia 1192 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840   ifcif 3882   {csn 3969   class class class wbr 4392    X. cxp 4938   ` cfv 5523  (class class class)co 6232   RRcr 9439   0cc0 9440    x. cmul 9445    <_ cle 9577   abscabs 13121   ~Hchil 26131   normhcno 26135   normfncnmf 26163   LinFnclf 26166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-hilex 26211  ax-hfvadd 26212  ax-hv0cl 26215  ax-hvaddid 26216  ax-hfvmul 26217  ax-hvmulid 26218  ax-hvmul0 26222  ax-hfi 26291  ax-his1 26294  ax-his3 26296  ax-his4 26297
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-2nd 6737  df-recs 6997  df-rdg 7031  df-er 7266  df-map 7377  df-en 7473  df-dom 7474  df-sdom 7475  df-sup 7853  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-n0 10755  df-z 10824  df-uz 11044  df-rp 11182  df-seq 12060  df-exp 12119  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-hnorm 26180  df-nmfn 27058  df-lnfn 27061
This theorem is referenced by:  lnfncnbd  27270
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