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Theorem 0cnfn 23483
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 9084 . . 3  |-  0  e.  CC
21fconst6 5633 . 2  |-  ( ~H 
X.  { 0 } ) : ~H --> CC
3 1rp 10616 . . . 4  |-  1  e.  RR+
4 c0ex 9085 . . . . . . . . . . . . 13  |-  0  e.  _V
54fvconst2 5947 . . . . . . . . . . . 12  |-  ( w  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  w )  =  0 )
64fvconst2 5947 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  x )  =  0 )
75, 6oveqan12rd 6101 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) )  =  ( 0  -  0 ) )
87adantlr 696 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  ( 0  -  0 ) )
91subidi 9371 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
108, 9syl6eq 2484 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  0 )
1110fveq2d 5732 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  ( abs `  0 ) )
12 abs0 12090 . . . . . . . 8  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2484 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  0 )
14 rpgt0 10623 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
1514ad2antlr 708 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  0  <  y
)
1613, 15eqbrtrd 4232 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )
1716a1d 23 . . . . 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 2789 . . . 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 4216 . . . . . . 7  |-  ( z  =  1  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  1 ) )
2019imbi1d 309 . . . . . 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 2725 . . . . 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 3052 . . . 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 645 . . 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 2802 . 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 23385 . 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 887 1  |-  ( ~H 
X.  { 0 } )  e.  ConFn
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   {csn 3814   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    < clt 9120    - cmin 9291   RR+crp 10612   abscabs 12039   ~Hchil 22422   normhcno 22426    -h cmv 22428   ConFnccnfn 22456
This theorem is referenced by:  nmcfnex  23556  nmcfnlb  23557  riesz4  23567  riesz1  23568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-hilex 22502
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-seq 11324  df-exp 11383  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-cnfn 23350
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