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Theorem kbass2 22697
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 5883 . . . 4  |-  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
)  e.  _V
2 eqid 2283 . . . 4  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
31, 2fnmpti 5372 . . 3  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  Fn  ~H
4 bracl 22529 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
5 brafn 22527 . . . . . 6  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
6 hfmmval 22319 . . . . . 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 463 . . . . 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 1146 . . . 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 5335 . . 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 224 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( ( bra `  A
) `  B )  .fn  ( bra `  C
) )  Fn  ~H )
11 brafn 22527 . . . . 5  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
12 kbop 22533 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
13 fco 5398 . . . . 5  |-  ( ( ( bra `  A
) : ~H --> CC  /\  ( B  ketbra  C ) : ~H --> ~H )  ->  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
1411, 12, 13syl2an 463 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( bra `  A )  o.  ( B  ketbra  C ) ) : ~H --> CC )
15143impb 1147 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
16 ffn 5389 . . 3  |-  ( ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) )  Fn 
~H )
1715, 16syl 15 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) )  Fn 
~H )
18 simpl1 958 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  ~H )
19 simpl2 959 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
20 braval 22524 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
2118, 19, 20syl2anc 642 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  =  ( B 
.ih  A ) )
22 simpl3 960 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
23 simpr 447 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
24 braval 22524 . . . . 5  |-  ( ( C  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  C
) `  x )  =  ( x  .ih  C ) )
2522, 23, 24syl2anc 642 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  C ) `  x
)  =  ( x 
.ih  C ) )
2621, 25oveq12d 5876 . . 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 21659 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
2819, 18, 27syl2anc 642 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( B  .ih  A )  e.  CC )
2921, 28eqeltrd 2357 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  e.  CC )
3022, 5syl 15 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( bra `  C
) : ~H --> CC )
31 hfmval 22324 . . . 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 1182 . . 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 21659 . . . . . 6  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
3423, 22, 33syl2anc 642 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
35 ax-his3 21663 . . . . 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 1182 . . . 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 973 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
38 fvco3 5596 . . . . . 6  |-  ( ( ( B  ketbra  C ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `
 x )  =  ( ( bra `  A
) `  ( ( B  ketbra  C ) `  x ) ) )
3937, 38sylan 457 . . . . 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 22534 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  x  e.  ~H )  ->  (
( B  ketbra  C ) `
 x )  =  ( ( x  .ih  C )  .h  B ) )
4119, 22, 23, 40syl3anc 1182 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( B 
ketbra  C ) `  x
)  =  ( ( x  .ih  C )  .h  B ) )
4241fveq2d 5529 . . . . 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 21593 . . . . . . 7  |-  ( ( ( x  .ih  C
)  e.  CC  /\  B  e.  ~H )  ->  ( ( x  .ih  C )  .h  B )  e.  ~H )
4434, 19, 43syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  .h  B )  e.  ~H )
45 braval 22524 . . . . . 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 642 . . . . 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 2319 . . . 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 8856 . . . 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 2325 . . 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 2325 . 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 5625 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   ~Hchil 21499    .h csm 21501    .ih csp 21502    .fn chft 21522   bracbr 21536    ketbra ck 21537
This theorem is referenced by:  kbass6  22701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-mulcom 8801  ax-hilex 21579  ax-hfvmul 21585  ax-hfi 21658  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-hfmul 22314  df-bra 22430  df-kb 22431
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