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Theorem kbass3 22714
Description: Dirac bra-ket associative law  <. A  |  B >.  <. C  |  D >.  =  ( <. A  |  B >.  <. C  |  )  |  D >.. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( bra `  A
) `  B )  x.  ( ( bra `  C
) `  D )
)  =  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  D
) )

Proof of Theorem kbass3
StepHypRef Expression
1 bracl 22545 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
21adantr 451 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( bra `  A ) `  B )  e.  CC )
3 brafn 22543 . . . 4  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
43ad2antrl 708 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( bra `  C ) : ~H --> CC )
5 simprr 733 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  D  e.  ~H )
6 hfmval 22340 . . 3  |-  ( ( ( ( bra `  A
) `  B )  e.  CC  /\  ( bra `  C ) : ~H --> CC  /\  D  e.  ~H )  ->  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  D
)  =  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  D )
) )
72, 4, 5, 6syl3anc 1182 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  D
)  =  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  D )
) )
87eqcomd 2301 1  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( bra `  A
) `  B )  x.  ( ( bra `  C
) `  D )
)  =  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  D
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   ~Hchil 21515    .fn chft 21538   bracbr 21552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-hilex 21595  ax-hfi 21674
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-hfmul 22330  df-bra 22446
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