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Theorem 0cnfn 27468
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 9634 . . 3  |-  0  e.  CC
21fconst6 5790 . 2  |-  ( ~H 
X.  { 0 } ) : ~H --> CC
3 1rp 11306 . . . 4  |-  1  e.  RR+
4 c0ex 9636 . . . . . . . . . . . . 13  |-  0  e.  _V
54fvconst2 6135 . . . . . . . . . . . 12  |-  ( w  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  w )  =  0 )
64fvconst2 6135 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  x )  =  0 )
75, 6oveqan12rd 6325 . . . . . . . . . . 11  |-  ( ( x  e.  ~H  /\  w  e.  ~H )  ->  ( ( ( ~H 
X.  { 0 } ) `  w )  -  ( ( ~H 
X.  { 0 } ) `  x ) )  =  ( 0  -  0 ) )
87adantlr 719 . . . . . . . . . 10  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  ( 0  -  0 ) )
9 0m0e0 10719 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
108, 9syl6eq 2486 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  0 )
1110fveq2d 5885 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  ( abs `  0 ) )
12 abs0 13327 . . . . . . . 8  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2486 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  0 )
14 rpgt0 11313 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
1514ad2antlr 731 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  0  <  y
)
1613, 15eqbrtrd 4446 . . . . . 6  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  <  y )
1716a1d 26 . . . . 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 2846 . . . 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 4430 . . . . . . 7  |-  ( z  =  1  ->  (
( normh `  ( w  -h  x ) )  < 
z  <->  ( normh `  (
w  -h  x ) )  <  1 ) )
2019imbi1d 318 . . . . . 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 2871 . . . . 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 3188 . . . 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 667 . . 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 2857 . 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 27370 . 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 928 1  |-  ( ~H 
X.  { 0 } )  e.  ConFn
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   {csn 4002   class class class wbr 4426    X. cxp 4852   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    < clt 9674    - cmin 9859   RR+crp 11302   abscabs 13276   ~Hchil 26407   normhcno 26411    -h cmv 26413   ConFnccnfn 26441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-hilex 26487
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-cnfn 27335
This theorem is referenced by:  nmcfnex  27541  nmcfnlb  27542  riesz4  27552  riesz1  27553
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