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Theorem bralnfn 26845
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 26844 . 2  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
2 simpll 753 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  A  e.  ~H )
3 hvmulcl 25908 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  .h  y
)  e.  ~H )
43ad2ant2lr 747 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
x  .h  y )  e.  ~H )
5 simprr 757 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  z  e.  ~H )
6 braadd 26842 . . . . . 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 1229 . . . . 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 26843 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  x  e.  CC  /\  y  e.  ~H )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
983expa 1197 . . . . . . 7  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  y  e.  ~H )  ->  ( ( bra `  A ) `  (
x  .h  y ) )  =  ( x  x.  ( ( bra `  A ) `  y
) ) )
109adantrr 716 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
1110oveq1d 6296 . . . . 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 2484 . . . 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 2864 . . 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 2857 . 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 26780 . 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 664 1  |-  ( A  e.  ~H  ->  ( bra `  A )  e. 
LinFn )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   -->wf 5574   ` cfv 5578  (class class class)co 6281   CCcc 9493    + caddc 9498    x. cmul 9500   ~Hchil 25814    +h cva 25815    .h csm 25816   LinFnclf 25849   bracbr 25851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-hilex 25894  ax-hfvadd 25895  ax-hfvmul 25900  ax-hfi 25974  ax-his2 25978  ax-his3 25979
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-map 7424  df-lnfn 26745  df-bra 26747
This theorem is referenced by:  rnbra  27004  kbass4  27016
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