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

Theorem kbmul 23463
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 22521 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
213adant3 978 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
3 simp3 960 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  C  e.  ~H )
4 kbfval 23460 . . 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 644 . 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 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  B  e.  ~H )
7 cjcl 11915 . . . . . 6  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
873ad2ant1 979 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  A )  e.  CC )
9 hvmulcl 22521 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  C  e.  ~H )  ->  ( ( * `  A )  .h  C
)  e.  ~H )
108, 3, 9syl2anc 644 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  .h  C )  e.  ~H )
11 kbfval 23460 . . . 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 644 . . 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 449 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
14 simpl3 963 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
15 hicl 22587 . . . . . . 7  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
1613, 14, 15syl2anc 644 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
17 simpl1 961 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
18 simpl2 962 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
19 ax-hvmulass 22515 . . . . . 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 1185 . . . . 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 9114 . . . . . . 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 22594 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  C  e.  ~H )  ->  (
x  .ih  ( (
* `  A )  .h  C ) )  =  ( A  x.  (
x  .ih  C )
) )
2317, 13, 14, 22syl3anc 1185 . . . . . . 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 2473 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  x.  A )  =  ( x  .ih  ( ( * `  A )  .h  C ) ) )
2524oveq1d 6099 . . . . 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 2472 . . . 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 4298 . . 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 2473 . 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 2473 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   CCcc 8993    x. cmul 9000   *ccj 11906   ~Hchil 22427    .h csm 22429    .ih csp 22430    ketbra ck 22465
This theorem is referenced by:  kbass6  23629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-hilex 22507  ax-hfvmul 22513  ax-hvmulass 22515  ax-hfi 22586  ax-his1 22589  ax-his3 22591
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-2 10063  df-cj 11909  df-re 11910  df-im 11911  df-kb 23359
  Copyright terms: Public domain W3C validator