HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  0cnfn Unicode version

Theorem 0cnfn 22576
Description: The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
0cnfn  |-  ( ~H 
X.  { 0 } )  e.  ConFn

Proof of Theorem 0cnfn
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0cn 8847 . . 3  |-  0  e.  CC
21fconst6 5447 . 2  |-  ( ~H 
X.  { 0 } ) : ~H --> CC
3 1rp 10374 . . . 4  |-  1  e.  RR+
4 c0ex 8848 . . . . . . . . . . . . 13  |-  0  e.  _V
54fvconst2 5745 . . . . . . . . . . . 12  |-  ( w  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  w )  =  0 )
64fvconst2 5745 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  x )  =  0 )
75, 6oveqan12rd 5894 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) )  =  ( 0  -  0 ) )
87adantlr 695 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  ( 0  -  0 ) )
91subidi 9133 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
108, 9syl6eq 2344 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  0 )
1110fveq2d 5545 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  ( abs `  0 ) )
12 abs0 11786 . . . . . . . 8  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2344 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  0 )
14 rpgt0 10381 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
1514ad2antlr 707 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  0  <  y
)
1613, 15eqbrtrd 4059 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )
1716a1d 22 . . . . 5  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( normh `  ( w  -h  x
) )  <  1  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) )
1817ralrimiva 2639 . . . 4  |-  ( ( x  e.  ~H  /\  y  e.  RR+ )  ->  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  1  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) )
19 breq2 4043 . . . . . . 7  |-  ( z  =  1  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  1 ) )
2019imbi1d 308 . . . . . 6  |-  ( z  =  1  ->  (
( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )  <-> 
( ( normh `  (
w  -h  x ) )  <  1  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) ) )
2120ralbidv 2576 . . . . 5  |-  ( z  =  1  ->  ( A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )  <->  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  1  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) ) )
2221rspcev 2897 . . . 4  |-  ( ( 1  e.  RR+  /\  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  <  1  ->  ( abs `  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) ) )  <  y
) )  ->  E. z  e.  RR+  A. w  e. 
~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) )
233, 18, 22sylancr 644 . . 3  |-  ( ( x  e.  ~H  /\  y  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  ( w  -h  x
) )  <  z  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) )
2423rgen2 2652 . 2  |-  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  (
( normh `  ( w  -h  x ) )  < 
z  ->  ( abs `  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) ) )  <  y
)
25 elcnfn 22478 . 2  |-  ( ( ~H  X.  { 0 } )  e.  ConFn  <->  (
( ~H  X.  {
0 } ) : ~H --> CC  /\  A. x  e.  ~H  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  ~H  ( ( normh `  (
w  -h  x ) )  <  z  -> 
( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y ) ) )
262, 24, 25mpbir2an 886 1  |-  ( ~H 
X.  { 0 } )  e.  ConFn
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {csn 3653   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    < clt 8883    - cmin 9053   RR+crp 10370   abscabs 11735   ~Hchil 21515   normhcno 21519    -h cmv 21521   ConFnccnfn 21549
This theorem is referenced by:  nmcfnex  22649  nmcfnlb  22650  riesz4  22660  riesz1  22661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hilex 21595
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-cnfn 22443
  Copyright terms: Public domain W3C validator