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Theorem kbass2 22713
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 5899 . . . 4  |-  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
)  e.  _V
2 eqid 2296 . . . 4  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
31, 2fnmpti 5388 . . 3  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  Fn  ~H
4 bracl 22545 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
5 brafn 22543 . . . . . 6  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
6 hfmmval 22335 . . . . . 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 5351 . . 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 22543 . . . . 5  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
12 kbop 22549 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
13 fco 5414 . . . . 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 5405 . . 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 22540 . . . . 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 22540 . . . . 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 5892 . . 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 21675 . . . . . 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 2370 . . . 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 22340 . . . 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 21675 . . . . . 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 21679 . . . . 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 5612 . . . . . 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 22550 . . . . . . 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 5545 . . . . 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 21609 . . . . . . 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 22540 . . . . . 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 2332 . . . 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 8872 . . . 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 2338 . . 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 2338 . 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 5641 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 1632    e. wcel 1696    e. cmpt 4093    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751    x. cmul 8758   ~Hchil 21515    .h csm 21517    .ih csp 21518    .fn chft 21538   bracbr 21552    ketbra ck 21553
This theorem is referenced by:  kbass6  22717
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-mulcom 8817  ax-hilex 21595  ax-hfvmul 21601  ax-hfi 21674  ax-his3 21679
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  df-kb 22447
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