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

Theorem bralnfn 22528
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 22527 . 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 21593 . . . . . . 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 22525 . . . . . 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 22526 . . . . . . . 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 5873 . . . . 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 2315 . . . 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 2635 . . 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 2626 . 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 22463 . 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 1623    e. wcel 1684   A.wral 2543   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742   ~Hchil 21499    +h cva 21500    .h csm 21501   LinFnclf 21534   bracbr 21536
This theorem is referenced by:  rnbra  22687  kbass4  22699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-hilex 21579  ax-hfvadd 21580  ax-hfvmul 21585  ax-hfi 21658  ax-his2 21662  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lnfn 22428  df-bra 22430
  Copyright terms: Public domain W3C validator