HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  kbmul Unicode version

Theorem kbmul 22649
Description: Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbmul  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  B
)  ketbra  C )  =  ( B  ketbra  ( ( * `  A )  .h  C ) ) )

Proof of Theorem kbmul
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hvmulcl 21707 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
213adant3 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
3 simp3 957 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  C  e.  ~H )
4 kbfval 22646 . . 3  |-  ( ( ( A  .h  B
)  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  .h  B )  ketbra  C )  =  ( x  e. 
~H  |->  ( ( x 
.ih  C )  .h  ( A  .h  B
) ) ) )
52, 3, 4syl2anc 642 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  B
)  ketbra  C )  =  ( x  e.  ~H  |->  ( ( x  .ih  C )  .h  ( A  .h  B ) ) ) )
6 simp2 956 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  B  e.  ~H )
7 cjcl 11686 . . . . . 6  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
873ad2ant1 976 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  A )  e.  CC )
9 hvmulcl 21707 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  C  e.  ~H )  ->  ( ( * `  A )  .h  C
)  e.  ~H )
108, 3, 9syl2anc 642 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  .h  C )  e.  ~H )
11 kbfval 22646 . . . 4  |-  ( ( B  e.  ~H  /\  ( ( * `  A )  .h  C
)  e.  ~H )  ->  ( B  ketbra  ( ( * `  A )  .h  C ) )  =  ( x  e. 
~H  |->  ( ( x 
.ih  ( ( * `
 A )  .h  C ) )  .h  B ) ) )
126, 10, 11syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  ( ( * `
 A )  .h  C ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  ( ( * `  A )  .h  C
) )  .h  B
) ) )
13 simpr 447 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
14 simpl3 960 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
15 hicl 21773 . . . . . . 7  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
1613, 14, 15syl2anc 642 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
17 simpl1 958 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
18 simpl2 959 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
19 ax-hvmulass 21701 . . . . . 6  |-  ( ( ( x  .ih  C
)  e.  CC  /\  A  e.  CC  /\  B  e.  ~H )  ->  (
( ( x  .ih  C )  x.  A )  .h  B )  =  ( ( x  .ih  C )  .h  ( A  .h  B ) ) )
2016, 17, 18, 19syl3anc 1182 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( x  .ih  C )  x.  A )  .h  B )  =  ( ( x  .ih  C
)  .h  ( A  .h  B ) ) )
2116, 17mulcomd 8946 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  x.  A )  =  ( A  x.  ( x 
.ih  C ) ) )
22 his52 21780 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  C  e.  ~H )  ->  (
x  .ih  ( (
* `  A )  .h  C ) )  =  ( A  x.  (
x  .ih  C )
) )
2317, 13, 14, 22syl3anc 1182 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  ( ( * `  A )  .h  C
) )  =  ( A  x.  ( x 
.ih  C ) ) )
2421, 23eqtr4d 2393 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  x.  A )  =  ( x  .ih  ( ( * `  A )  .h  C ) ) )
2524oveq1d 5960 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( ( x  .ih  C )  x.  A )  .h  B )  =  ( ( x  .ih  (
( * `  A
)  .h  C ) )  .h  B ) )
2620, 25eqtr3d 2392 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  .h  ( A  .h  B
) )  =  ( ( x  .ih  (
( * `  A
)  .h  C ) )  .h  B ) )
2726mpteq2dva 4187 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
x  e.  ~H  |->  ( ( x  .ih  C
)  .h  ( A  .h  B ) ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  ( ( * `  A )  .h  C ) )  .h  B ) ) )
2812, 27eqtr4d 2393 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  ketbra  ( ( * `
 A )  .h  C ) )  =  ( x  e.  ~H  |->  ( ( x  .ih  C )  .h  ( A  .h  B ) ) ) )
295, 28eqtr4d 2393 1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  .h  B
)  ketbra  C )  =  ( B  ketbra  ( ( * `  A )  .h  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   CCcc 8825    x. cmul 8832   *ccj 11677   ~Hchil 21613    .h csm 21615    .ih csp 21616    ketbra ck 21651
This theorem is referenced by:  kbass6  22815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-hilex 21693  ax-hfvmul 21699  ax-hvmulass 21701  ax-hfi 21772  ax-his1 21775  ax-his3 21777
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-2 9894  df-cj 11680  df-re 11681  df-im 11682  df-kb 22545
  Copyright terms: Public domain W3C validator