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

Theorem 0cnfn 25389
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 9383 . . 3  |-  0  e.  CC
21fconst6 5605 . 2  |-  ( ~H 
X.  { 0 } ) : ~H --> CC
3 1rp 11000 . . . 4  |-  1  e.  RR+
4 c0ex 9385 . . . . . . . . . . . . 13  |-  0  e.  _V
54fvconst2 5938 . . . . . . . . . . . 12  |-  ( w  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  w )  =  0 )
64fvconst2 5938 . . . . . . . . . . . 12  |-  ( x  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  x )  =  0 )
75, 6oveqan12rd 6116 . . . . . . . . . . 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 10436 . . . . . . . . . 10  |-  ( 0  -  0 )  =  0
108, 9syl6eq 2491 . . . . . . . . 9  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( ( ( ~H  X.  { 0 } ) `  w
)  -  ( ( ~H  X.  { 0 } ) `  x
) )  =  0 )
1110fveq2d 5700 . . . . . . . 8  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  ( abs `  0 ) )
12 abs0 12779 . . . . . . . 8  |-  ( abs `  0 )  =  0
1311, 12syl6eq 2491 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  ( abs `  (
( ( ~H  X.  { 0 } ) `
 w )  -  ( ( ~H  X.  { 0 } ) `
 x ) ) )  =  0 )
14 rpgt0 11007 . . . . . . . 8  |-  ( y  e.  RR+  ->  0  < 
y )
1514ad2antlr 726 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  RR+ )  /\  w  e.  ~H )  ->  0  <  y
)
1613, 15eqbrtrd 4317 . . . . . 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 2804 . . . 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 4301 . . . . . . 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 2740 . . . . 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 3078 . . . 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 2817 . 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 25291 . 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 911 1  |-  ( ~H 
X.  { 0 } )  e.  ConFn
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {csn 3882   class class class wbr 4297    X. cxp 4843   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288    < clt 9423    - cmin 9600   RR+crp 10996   abscabs 12728   ~Hchil 24326   normhcno 24330    -h cmv 24332   ConFnccnfn 24360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-hilex 24406
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-seq 11812  df-exp 11871  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-cnfn 25256
This theorem is referenced by:  nmcfnex  25462  nmcfnlb  25463  riesz4  25473  riesz1  25474
  Copyright terms: Public domain W3C validator