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

Theorem bralnfn 22544
Description: The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
bralnfn  |-  ( A  e.  ~H  ->  ( bra `  A )  e. 
LinFn )

Proof of Theorem bralnfn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brafn 22543 . 2  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
2 simpll 730 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  A  e.  ~H )
3 hvmulcl 21609 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  .h  y
)  e.  ~H )
43ad2ant2lr 728 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
x  .h  y )  e.  ~H )
5 simprr 733 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  z  e.  ~H )
6 braadd 22541 . . . . . 6  |-  ( ( A  e.  ~H  /\  ( x  .h  y
)  e.  ~H  /\  z  e.  ~H )  ->  ( ( bra `  A
) `  ( (
x  .h  y )  +h  z ) )  =  ( ( ( bra `  A ) `
 ( x  .h  y ) )  +  ( ( bra `  A
) `  z )
) )
72, 4, 5, 6syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( bra `  A
) `  ( (
x  .h  y )  +h  z ) )  =  ( ( ( bra `  A ) `
 ( x  .h  y ) )  +  ( ( bra `  A
) `  z )
) )
8 bramul 22542 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  x  e.  CC  /\  y  e.  ~H )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
983expa 1151 . . . . . . 7  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  y  e.  ~H )  ->  ( ( bra `  A ) `  (
x  .h  y ) )  =  ( x  x.  ( ( bra `  A ) `  y
) ) )
109adantrr 697 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
1110oveq1d 5889 . . . . 5  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( ( bra `  A
) `  ( x  .h  y ) )  +  ( ( bra `  A
) `  z )
)  =  ( ( x  x.  ( ( bra `  A ) `
 y ) )  +  ( ( bra `  A ) `  z
) ) )
127, 11eqtrd 2328 . . . 4  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( bra `  A
) `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( bra `  A ) `  y
) )  +  ( ( bra `  A
) `  z )
) )
1312ralrimivva 2648 . . 3  |-  ( ( A  e.  ~H  /\  x  e.  CC )  ->  A. y  e.  ~H  A. z  e.  ~H  (
( bra `  A
) `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( bra `  A ) `  y
) )  +  ( ( bra `  A
) `  z )
) )
1413ralrimiva 2639 . 2  |-  ( A  e.  ~H  ->  A. x  e.  CC  A. y  e. 
~H  A. z  e.  ~H  ( ( bra `  A
) `  ( (
x  .h  y )  +h  z ) )  =  ( ( x  x.  ( ( bra `  A ) `  y
) )  +  ( ( bra `  A
) `  z )
) )
15 ellnfn 22479 . 2  |-  ( ( bra `  A )  e.  LinFn 
<->  ( ( bra `  A
) : ~H --> CC  /\  A. x  e.  CC  A. y  e.  ~H  A. z  e.  ~H  ( ( bra `  A ) `  (
( x  .h  y
)  +h  z ) )  =  ( ( x  x.  ( ( bra `  A ) `
 y ) )  +  ( ( bra `  A ) `  z
) ) ) )
161, 14, 15sylanbrc 645 1  |-  ( A  e.  ~H  ->  ( bra `  A )  e. 
LinFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758   ~Hchil 21515    +h cva 21516    .h csm 21517   LinFnclf 21550   bracbr 21552
This theorem is referenced by:  rnbra  22703  kbass4  22715
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-hilex 21595  ax-hfvadd 21596  ax-hfvmul 21601  ax-hfi 21674  ax-his2 21678  ax-his3 21679
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-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-map 6790  df-lnfn 22444  df-bra 22446
  Copyright terms: Public domain W3C validator