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Theorem kbass2 27234
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 6298 . . . 4  |-  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
)  e.  _V
2 eqid 2454 . . . 4  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  =  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )
31, 2fnmpti 5691 . . 3  |-  ( x  e.  ~H  |->  ( ( ( bra `  A
) `  B )  x.  ( ( bra `  C
) `  x )
) )  Fn  ~H
4 bracl 27066 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  e.  CC )
5 brafn 27064 . . . . . 6  |-  ( C  e.  ~H  ->  ( bra `  C ) : ~H --> CC )
6 hfmmval 26856 . . . . . 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 475 . . . . 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 1189 . . . 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 5653 . . 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 27064 . . . . 5  |-  ( A  e.  ~H  ->  ( bra `  A ) : ~H --> CC )
12 kbop 27070 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
13 fco 5723 . . . . 5  |-  ( ( ( bra `  A
) : ~H --> CC  /\  ( B  ketbra  C ) : ~H --> ~H )  ->  ( ( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
1411, 12, 13syl2an 475 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( bra `  A )  o.  ( B  ketbra  C ) ) : ~H --> CC )
15143impb 1190 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( bra `  A
)  o.  ( B 
ketbra  C ) ) : ~H --> CC )
16 ffn 5713 . . 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 997 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  ~H )
19 simpl2 998 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
20 braval 27061 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( bra `  A
) `  B )  =  ( B  .ih  A ) )
2118, 19, 20syl2anc 659 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  A ) `  B
)  =  ( B 
.ih  A ) )
22 simpl3 999 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
23 simpr 459 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
24 braval 27061 . . . . 5  |-  ( ( C  e.  ~H  /\  x  e.  ~H )  ->  ( ( bra `  C
) `  x )  =  ( x  .ih  C ) )
2522, 23, 24syl2anc 659 . . . 4  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( bra `  C ) `  x
)  =  ( x 
.ih  C ) )
2621, 25oveq12d 6288 . . 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 26195 . . . . . 6  |-  ( ( B  e.  ~H  /\  A  e.  ~H )  ->  ( B  .ih  A
)  e.  CC )
2819, 18, 27syl2anc 659 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( B  .ih  A )  e.  CC )
2921, 28eqeltrd 2542 . . . 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 26861 . . . 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 1226 . . 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 26195 . . . . . 6  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
3423, 22, 33syl2anc 659 . . . . 5  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
35 ax-his3 26199 . . . . 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 1226 . . . 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 1012 . . . . . 6  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  C ) : ~H --> ~H )
38 fvco3 5925 . . . . . 6  |-  ( ( ( B  ketbra  C ) : ~H --> ~H  /\  x  e.  ~H )  ->  ( ( ( bra `  A )  o.  ( B  ketbra  C ) ) `
 x )  =  ( ( bra `  A
) `  ( ( B  ketbra  C ) `  x ) ) )
3937, 38sylan 469 . . . . 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 27071 . . . . . . 7  |-  ( ( B  e.  ~H  /\  C  e.  ~H  /\  x  e.  ~H )  ->  (
( B  ketbra  C ) `
 x )  =  ( ( x  .ih  C )  .h  B ) )
4119, 22, 23, 40syl3anc 1226 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( B 
ketbra  C ) `  x
)  =  ( ( x  .ih  C )  .h  B ) )
4241fveq2d 5852 . . . . 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 26128 . . . . . . 7  |-  ( ( ( x  .ih  C
)  e.  CC  /\  B  e.  ~H )  ->  ( ( x  .ih  C )  .h  B )  e.  ~H )
4434, 19, 43syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  .h  B )  e.  ~H )
45 braval 27061 . . . . . 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 659 . . . . 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 2499 . . . 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 9606 . . . 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 2505 . . 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 2505 . 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 5960 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    |-> cmpt 4497    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479    x. cmul 9486   ~Hchil 26034    .h csm 26036    .ih csp 26037    .fn chft 26057   bracbr 26071    ketbra ck 26072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-mulcom 9545  ax-hilex 26114  ax-hfvmul 26120  ax-hfi 26194  ax-his3 26199
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-hfmul 26851  df-bra 26967  df-kb 26968
This theorem is referenced by:  kbass6  27238
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