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

Theorem 0lnfn 22581
Description: The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)
Assertion
Ref Expression
0lnfn  |-  ( ~H 
X.  { 0 } )  e.  LinFn

Proof of Theorem 0lnfn
Dummy variables  x  y  z 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 hvmulcl 21609 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  .h  y
)  e.  ~H )
4 hvaddcl 21608 . . . . . . 7  |-  ( ( ( x  .h  y
)  e.  ~H  /\  z  e.  ~H )  ->  ( ( x  .h  y )  +h  z
)  e.  ~H )
53, 4sylan 457 . . . . . 6  |-  ( ( ( x  e.  CC  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  .h  y )  +h  z )  e.  ~H )
6 c0ex 8848 . . . . . . 7  |-  0  e.  _V
76fvconst2 5745 . . . . . 6  |-  ( ( ( x  .h  y
)  +h  z )  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  ( ( x  .h  y )  +h  z
) )  =  0 )
85, 7syl 15 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( ~H 
X.  { 0 } ) `  ( ( x  .h  y )  +h  z ) )  =  0 )
96fvconst2 5745 . . . . . . . . 9  |-  ( y  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  y )  =  0 )
109oveq2d 5890 . . . . . . . 8  |-  ( y  e.  ~H  ->  (
x  x.  ( ( ~H  X.  { 0 } ) `  y
) )  =  ( x  x.  0 ) )
11 mul01 9007 . . . . . . . 8  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
1210, 11sylan9eqr 2350 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  x.  (
( ~H  X.  {
0 } ) `  y ) )  =  0 )
136fvconst2 5745 . . . . . . 7  |-  ( z  e.  ~H  ->  (
( ~H  X.  {
0 } ) `  z )  =  0 )
1412, 13oveqan12d 5893 . . . . . 6  |-  ( ( ( x  e.  CC  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  x.  ( ( ~H 
X.  { 0 } ) `  y ) )  +  ( ( ~H  X.  { 0 } ) `  z
) )  =  ( 0  +  0 ) )
15 00id 9003 . . . . . 6  |-  ( 0  +  0 )  =  0
1614, 15syl6eq 2344 . . . . 5  |-  ( ( ( x  e.  CC  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( x  x.  ( ( ~H 
X.  { 0 } ) `  y ) )  +  ( ( ~H  X.  { 0 } ) `  z
) )  =  0 )
178, 16eqtr4d 2331 . . . 4  |-  ( ( ( x  e.  CC  /\  y  e.  ~H )  /\  z  e.  ~H )  ->  ( ( ~H 
X.  { 0 } ) `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( ~H 
X.  { 0 } ) `  y ) )  +  ( ( ~H  X.  { 0 } ) `  z
) ) )
18173impa 1146 . . 3  |-  ( ( x  e.  CC  /\  y  e.  ~H  /\  z  e.  ~H )  ->  (
( ~H  X.  {
0 } ) `  ( ( x  .h  y )  +h  z
) )  =  ( ( x  x.  (
( ~H  X.  {
0 } ) `  y ) )  +  ( ( ~H  X.  { 0 } ) `
 z ) ) )
1918rgen3 2653 . 2  |-  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( ( ~H  X.  { 0 } ) `
 ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( ~H  X.  { 0 } ) `
 y ) )  +  ( ( ~H 
X.  { 0 } ) `  z ) )
20 ellnfn 22479 . 2  |-  ( ( ~H  X.  { 0 } )  e.  LinFn  <->  (
( ~H  X.  {
0 } ) : ~H --> CC  /\  A. x  e.  CC  A. y  e.  ~H  A. z  e. 
~H  ( ( ~H 
X.  { 0 } ) `  ( ( x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( ~H 
X.  { 0 } ) `  y ) )  +  ( ( ~H  X.  { 0 } ) `  z
) ) ) )
212, 19, 20mpbir2an 886 1  |-  ( ~H 
X.  { 0 } )  e.  LinFn
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    + caddc 8756    x. cmul 8758   ~Hchil 21515    +h cva 21516    .h csm 21517   LinFnclf 21550
This theorem is referenced by:  nmfn0  22583  lnfn0  22643  lnfnmul  22644  nmbdfnlb  22646  nmcfnex  22649  nmcfnlb  22650  lnfncon  22652  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-hilex 21595  ax-hfvadd 21596  ax-hfvmul 21601
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-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  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-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-lnfn 22444
  Copyright terms: Public domain W3C validator