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Theorem kbass2 25456
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 6115 . . . 4  |-  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
)  e.  _V
2 eqid 2441 . . . 4  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
31, 2fnmpti 5536 . . 3  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  Fn  ~H
4 bracl 25288 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
5 brafn 25286 . . . . . 6  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
6 hfmmval 25078 . . . . . 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 474 . . . . 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 1177 . . . 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 5498 . . 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 233 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  Fn  ~H )
11 brafn 25286 . . . . 5  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
12 kbop 25292 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
13 fco 5565 . . . . 5  |-  ( ( ( bra `  A
) : ~H --> CC  /\  ( B  ketbra  C ) : ~H --> ~H )  ->  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
1411, 12, 13syl2an 474 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( bra `  A )  o.  ( B  ketbra  C ) ) : ~H --> CC )
15143impb 1178 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
16 ffn 5556 . . 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 986 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  ~H )
19 simpl2 987 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
20 braval 25283 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
2118, 19, 20syl2anc 656 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  =  ( B 
.ih  A ) )
22 simpl3 988 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
23 simpr 458 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
24 braval 25283 . . . . 5  |-  ( ( C  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  C
) `  x )  =  ( x  .ih  C ) )
2522, 23, 24syl2anc 656 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  C ) `  x
)  =  ( x 
.ih  C ) )
2621, 25oveq12d 6108 . . 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 24417 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
2819, 18, 27syl2anc 656 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( B  .ih  A )  e.  CC )
2921, 28eqeltrd 2515 . . . 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 25083 . . . 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 1213 . . 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 24417 . . . . . 6  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
3423, 22, 33syl2anc 656 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
35 ax-his3 24421 . . . . 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 1213 . . . 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 1001 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
38 fvco3 5765 . . . . . 6  |-  ( ( ( B  ketbra  C ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `
 x )  =  ( ( bra `  A
) `  ( ( B  ketbra  C ) `  x ) ) )
3937, 38sylan 468 . . . . 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 25293 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  x  e.  ~H )  ->  (
( B  ketbra  C ) `
 x )  =  ( ( x  .ih  C )  .h  B ) )
4119, 22, 23, 40syl3anc 1213 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( B 
ketbra  C ) `  x
)  =  ( ( x  .ih  C )  .h  B ) )
4241fveq2d 5692 . . . . 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 24350 . . . . . . 7  |-  ( ( ( x  .ih  C
)  e.  CC  /\  B  e.  ~H )  ->  ( ( x  .ih  C )  .h  B )  e.  ~H )
4434, 19, 43syl2anc 656 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  .h  B )  e.  ~H )
45 braval 25283 . . . . . 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 656 . . . . 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 2477 . . . 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 9403 . . . 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 2483 . . 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 2483 . 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 5797 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    e. cmpt 4347    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276    x. cmul 9283   ~Hchil 24256    .h csm 24258    .ih csp 24259    .fn chft 24279   bracbr 24293    ketbra ck 24294
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-mulcom 9342  ax-hilex 24336  ax-hfvmul 24342  ax-hfi 24416  ax-his3 24421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7212  df-hfmul 25073  df-bra 25189  df-kb 25190
This theorem is referenced by:  kbass6  25460
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