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Theorem kbass2 23620
Description: Dirac bra-ket associative law  ( <. A  |  B >. ) <. C  |  =  <. A  | 
(  |  B >.  <. C  |  ) i.e. the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  =  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) )

Proof of Theorem kbass2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ovex 6106 . . . 4  |-  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
)  e.  _V
2 eqid 2436 . . . 4  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
31, 2fnmpti 5573 . . 3  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  Fn  ~H
4 bracl 23452 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
5 brafn 23450 . . . . . 6  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
6 hfmmval 23242 . . . . . 6  |-  ( ( ( ( bra `  A
) `  B )  e.  CC  /\  ( bra `  C ) : ~H --> CC )  ->  ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) ) )
74, 5, 6syl2an 464 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  C  e.  ~H )  ->  ( ( ( bra `  A ) `
 B )  .fn  ( bra `  C ) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) ) )
873impa 1148 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) ) )
98fneq1d 5536 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( ( bra `  A ) `  B
)  .fn  ( bra `  C ) )  Fn 
~H 
<->  ( x  e.  ~H  |->  ( ( ( bra `  A ) `  B
)  x.  ( ( bra `  C ) `
 x ) ) )  Fn  ~H )
)
103, 9mpbiri 225 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  Fn  ~H )
11 brafn 23450 . . . . 5  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
12 kbop 23456 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
13 fco 5600 . . . . 5  |-  ( ( ( bra `  A
) : ~H --> CC  /\  ( B  ketbra  C ) : ~H --> ~H )  ->  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
1411, 12, 13syl2an 464 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( bra `  A )  o.  ( B  ketbra  C ) ) : ~H --> CC )
15143impb 1149 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
16 ffn 5591 . . 3  |-  ( ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) )  Fn 
~H )
1715, 16syl 16 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) )  Fn 
~H )
18 simpl1 960 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  ~H )
19 simpl2 961 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
20 braval 23447 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
2118, 19, 20syl2anc 643 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  =  ( B 
.ih  A ) )
22 simpl3 962 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
23 simpr 448 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
24 braval 23447 . . . . 5  |-  ( ( C  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  C
) `  x )  =  ( x  .ih  C ) )
2522, 23, 24syl2anc 643 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  C ) `  x
)  =  ( x 
.ih  C ) )
2621, 25oveq12d 6099 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( bra `  A ) `
 B )  x.  ( ( bra `  C
) `  x )
)  =  ( ( B  .ih  A )  x.  ( x  .ih  C ) ) )
27 hicl 22582 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
2819, 18, 27syl2anc 643 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( B  .ih  A )  e.  CC )
2921, 28eqeltrd 2510 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  e.  CC )
3022, 5syl 16 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( bra `  C
) : ~H --> CC )
31 hfmval 23247 . . . 4  |-  ( ( ( ( bra `  A
) `  B )  e.  CC  /\  ( bra `  C ) : ~H --> CC  /\  x  e.  ~H )  ->  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  x
)  =  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
3229, 30, 23, 31syl3anc 1184 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  x
)  =  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
33 hicl 22582 . . . . . 6  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
3423, 22, 33syl2anc 643 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
35 ax-his3 22586 . . . . 5  |-  ( ( ( x  .ih  C
)  e.  CC  /\  B  e.  ~H  /\  A  e.  ~H )  ->  (
( ( x  .ih  C )  .h  B ) 
.ih  A )  =  ( ( x  .ih  C )  x.  ( B 
.ih  A ) ) )
3634, 19, 18, 35syl3anc 1184 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( x  .ih  C )  .h  B )  .ih  A )  =  ( ( x  .ih  C )  x.  ( B  .ih  A ) ) )
37123adant1 975 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
38 fvco3 5800 . . . . . 6  |-  ( ( ( B  ketbra  C ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `
 x )  =  ( ( bra `  A
) `  ( ( B  ketbra  C ) `  x ) ) )
3937, 38sylan 458 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `  x
)  =  ( ( bra `  A ) `
 ( ( B 
ketbra  C ) `  x
) ) )
40 kbval 23457 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  x  e.  ~H )  ->  (
( B  ketbra  C ) `
 x )  =  ( ( x  .ih  C )  .h  B ) )
4119, 22, 23, 40syl3anc 1184 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( B 
ketbra  C ) `  x
)  =  ( ( x  .ih  C )  .h  B ) )
4241fveq2d 5732 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  (
( B  ketbra  C ) `
 x ) )  =  ( ( bra `  A ) `  (
( x  .ih  C
)  .h  B ) ) )
43 hvmulcl 22516 . . . . . . 7  |-  ( ( ( x  .ih  C
)  e.  CC  /\  B  e.  ~H )  ->  ( ( x  .ih  C )  .h  B )  e.  ~H )
4434, 19, 43syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  .h  B )  e.  ~H )
45 braval 23447 . . . . . 6  |-  ( ( A  e.  ~H  /\  ( ( x  .ih  C )  .h  B )  e.  ~H )  -> 
( ( bra `  A
) `  ( (
x  .ih  C )  .h  B ) )  =  ( ( ( x 
.ih  C )  .h  B )  .ih  A
) )
4618, 44, 45syl2anc 643 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  (
( x  .ih  C
)  .h  B ) )  =  ( ( ( x  .ih  C
)  .h  B ) 
.ih  A ) )
4739, 42, 463eqtrd 2472 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `  x
)  =  ( ( ( x  .ih  C
)  .h  B ) 
.ih  A ) )
4828, 34mulcomd 9109 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( B 
.ih  A )  x.  ( x  .ih  C
) )  =  ( ( x  .ih  C
)  x.  ( B 
.ih  A ) ) )
4936, 47, 483eqtr4d 2478 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `  x
)  =  ( ( B  .ih  A )  x.  ( x  .ih  C ) ) )
5026, 32, 493eqtr4d 2478 . 2  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( ( bra `  A
) `  B )  .fn  ( bra `  C
) ) `  x
)  =  ( ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) `  x ) )
5110, 17, 50eqfnfvd 5830 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  =  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    e. cmpt 4266    o. ccom 4882    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988    x. cmul 8995   ~Hchil 22422    .h csm 22424    .ih csp 22425    .fn chft 22445   bracbr 22459    ketbra ck 22460
This theorem is referenced by:  kbass6  23624
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-mulcom 9054  ax-hilex 22502  ax-hfvmul 22508  ax-hfi 22581  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-hfmul 23237  df-bra 23353  df-kb 23354
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