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Theorem brafval 27572
Description: The bra of a vector, expressed as  <. A  | in Dirac notation. See df-bra 27479. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
brafval  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
Distinct variable group:    x, A

Proof of Theorem brafval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oveq2 6305 . . 3  |-  ( y  =  A  ->  (
x  .ih  y )  =  ( x  .ih  A ) )
21mpteq2dv 4505 . 2  |-  ( y  =  A  ->  (
x  e.  ~H  |->  ( x  .ih  y ) )  =  ( x  e.  ~H  |->  ( x 
.ih  A ) ) )
3 df-bra 27479 . 2  |-  bra  =  ( y  e.  ~H  |->  ( x  e.  ~H  |->  ( x  .ih  y ) ) )
4 ax-hilex 26628 . . 3  |-  ~H  e.  _V
54mptex 6143 . 2  |-  ( x  e.  ~H  |->  ( x 
.ih  A ) )  e.  _V
62, 3, 5fvmpt 5956 1  |-  ( A  e.  ~H  ->  ( bra `  A )  =  ( x  e.  ~H  |->  ( x  .ih  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867    |-> cmpt 4476   ` cfv 5593  (class class class)co 6297   ~Hchil 26548    .ih csp 26551   bracbr 26585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4530  ax-sep 4540  ax-nul 4548  ax-pr 4653  ax-hilex 26628
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-bra 27479
This theorem is referenced by:  braval  27573  brafn  27576  bra0  27579  brafnmul  27580
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