Theorem bravalvalt 9784

Description: A bra-ket juxtaposition, expressed as <.A | B>. in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186.

Assertion

Ref	Expression
bravalvalt	|- ((A e. H~ /\ B e. H~) -> ((bra` A)` B) = (B .ih A))

Proof of Theorem bravalvalt

Step	Hyp	Ref	Expression
1		bravalt 9783	. . 3 |- (A e. H~ -> (bra` A) = {<.x, y>. | (x e. H~ /\ y = (x .ih A))})
2	1	fveq1d 3711	. 2 |- (A e. H~ -> ((bra` A)` B) = ({<.x, y>. | (x e. H~ /\ y = (x .ih A))}` B))
3		opreq1 3953	. . 3 |- (x = B -> (x .ih A) = (B .ih A))
4		eqid 1468	. . 3 |- {<.x, y>. | (x e. H~ /\ y = (x .ih A))} = {<.x, y>. | (x e. H~ /\ y = (x .ih A))}
5		oprex 3968	. . 3 |- (B .ih A) e. V
6	3, 4, 5	fvopab4 3765	. 2 |- (B e. H~ -> ({<.x, y>. | (x e. H~ /\ y = (x .ih A))}` B) = (B .ih A))
7	2, 6	sylan9eq 1519	1 |- ((A e. H~ /\ B e. H~) -> ((bra` A)` B) = (B .ih A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {copab 2656  ` cfv 3172  (class class class)co 3948  H~chil 8727   .ih csp 8732  bracbr 8764

This theorem is referenced by:  braaddt 9785  bramult 9786  brafnmult 9791  branmfnt 9951  bra11 9954  cnvbravalt 9956  kbass1t 9961  kbass2t 9962  kbass6t 9966

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-hilex 8790

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-opr 3950  df-bra 9693

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