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Theorem 0cnfn 26572
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 9584 . . 3  |-  0  e.  CC
21fconst6 5773 . 2  |-  ( ~H 
X.  { 0 } ) : ~H --> CC
3 1rp 11220 . . . 4  |-  1  e.  RR+
4 c0ex 9586 . . . . . . . . . . . . 13  |-  0  e.  _V
54fvconst2 6114 . . . . . . . . . . . 12  |-  ( w  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  w )  =  0 )
64fvconst2 6114 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  x )  =  0 )
75, 6oveqan12rd 6302 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) )  =  ( 0  -  0 ) )
87adantlr 714 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  ( 0  -  0 ) )
9 0m0e0 10641 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
108, 9syl6eq 2524 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  0 )
1110fveq2d 5868 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  ( abs `  0 ) )
12 abs0 13075 . . . . . . . 8  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2524 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  0 )
14 rpgt0 11227 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
1514ad2antlr 726 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  0  <  y
)
1613, 15eqbrtrd 4467 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )
1716a1d 25 . . . . 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 2878 . . . 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 4451 . . . . . . 7  |-  ( z  =  1  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  1 ) )
2019imbi1d 317 . . . . . 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 2903 . . . . 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 3214 . . . 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 663 . . 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 2889 . 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 26474 . 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 918 1  |-  ( ~H 
X.  { 0 } )  e.  ConFn
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {csn 4027   class class class wbr 4447    X. cxp 4997   -->wf 5582   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    < clt 9624    - cmin 9801   RR+crp 11216   abscabs 13024   ~Hchil 25509   normhcno 25513    -h cmv 25515   ConFnccnfn 25543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-hilex 25589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-seq 12071  df-exp 12130  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-cnfn 26439
This theorem is referenced by:  nmcfnex  26645  nmcfnlb  26646  riesz4  26656  riesz1  26657
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