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Theorem bralnfn 26531
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 26530 . 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 25594 . . . . . . 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 756 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  x  e.  CC )  /\  ( y  e. 
~H  /\  z  e.  ~H ) )  ->  z  e.  ~H )
6 braadd 26528 . . . . . 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 1223 . . . . 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 26529 . . . . . . . 8  |-  ( ( A  e.  ~H  /\  x  e.  CC  /\  y  e.  ~H )  ->  (
( bra `  A
) `  ( x  .h  y ) )  =  ( x  x.  (
( bra `  A
) `  y )
) )
983expa 1191 . . . . . . 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 6292 . . . . 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 2503 . . . 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 2880 . . 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 2873 . 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 26466 . 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 1374    e. wcel 1762   A.wral 2809   -->wf 5577   ` cfv 5581  (class class class)co 6277   CCcc 9481    + caddc 9486    x. cmul 9488   ~Hchil 25500    +h cva 25501    .h csm 25502   LinFnclf 25535   bracbr 25537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-hilex 25580  ax-hfvadd 25581  ax-hfvmul 25586  ax-hfi 25660  ax-his2 25664  ax-his3 25665
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-lnfn 26431  df-bra 26433
This theorem is referenced by:  rnbra  26690  kbass4  26702
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