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Theorem kbmul 23411
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 22469 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
213adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  B )  e.  ~H )
3 simp3 959 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  C  e.  ~H )
4 kbfval 23408 . . 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 643 . 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 958 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  B  e.  ~H )
7 cjcl 11865 . . . . . 6  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
873ad2ant1 978 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  A )  e.  CC )
9 hvmulcl 22469 . . . . 5  |-  ( ( ( * `  A
)  e.  CC  /\  C  e.  ~H )  ->  ( ( * `  A )  .h  C
)  e.  ~H )
108, 3, 9syl2anc 643 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( * `  A
)  .h  C )  e.  ~H )
11 kbfval 23408 . . . 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 643 . . 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 448 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  x  e.  ~H )
14 simpl3 962 . . . . . . 7  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  C  e.  ~H )
15 hicl 22535 . . . . . . 7  |-  ( ( x  e.  ~H  /\  C  e.  ~H )  ->  ( x  .ih  C
)  e.  CC )
1613, 14, 15syl2anc 643 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( x  .ih  C )  e.  CC )
17 simpl1 960 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  A  e.  CC )
18 simpl2 961 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  B  e.  ~H )
19 ax-hvmulass 22463 . . . . . 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 1184 . . . . 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 9065 . . . . . . 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 22542 . . . . . . . 8  |-  ( ( A  e.  CC  /\  x  e.  ~H  /\  C  e.  ~H )  ->  (
x  .ih  ( (
* `  A )  .h  C ) )  =  ( A  x.  (
x  .ih  C )
) )
2317, 13, 14, 22syl3anc 1184 . . . . . . 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 2439 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  x.  A )  =  ( x  .ih  ( ( * `  A )  .h  C ) ) )
2524oveq1d 6055 . . . . 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 2438 . . . 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 4255 . . 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 2439 . 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 2439 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944    x. cmul 8951   *ccj 11856   ~Hchil 22375    .h csm 22377    .ih csp 22378    ketbra ck 22413
This theorem is referenced by:  kbass6  23577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-hilex 22455  ax-hfvmul 22461  ax-hvmulass 22463  ax-hfi 22534  ax-his1 22537  ax-his3 22539
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-cj 11859  df-re 11860  df-im 11861  df-kb 23307
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