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Theorem bralnfn 23451
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 23450 . 2  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
2 simpll 731 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  A  e.  ~H )
3 hvmulcl 22516 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  .h  y
)  e.  ~H )
43ad2ant2lr 729 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
x  .h  y )  e.  ~H )
5 simprr 734 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  z  e.  ~H )
6 braadd 23448 . . . . . 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 1184 . . . . 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 23449 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  x  e.  CC  /\  y  e.  ~H )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
983expa 1153 . . . . . . 7  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  y  e.  ~H )  ->  ( ( bra `  A ) `  (
x  .h  y ) )  =  ( x  x.  ( ( bra `  A ) `  y
) ) )
109adantrr 698 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
1110oveq1d 6096 . . . . 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 2468 . . . 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 2798 . . 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 2789 . 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 23386 . 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 646 1  |-  ( A  e.  ~H  ->  ( bra `  A )  e. 
LinFn )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    + caddc 8993    x. cmul 8995   ~Hchil 22422    +h cva 22423    .h csm 22424   LinFnclf 22457   bracbr 22459
This theorem is referenced by:  rnbra  23610  kbass4  23622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-hilex 22502  ax-hfvadd 22503  ax-hfvmul 22508  ax-hfi 22581  ax-his2 22585  ax-his3 22586
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lnfn 23351  df-bra 23353
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