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Theorem kbmul 22535
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 21593 . . . 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 22532 . . 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 11590 . . . . . 6  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
873ad2ant1 976 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
* `  A )  e.  CC )
9 hvmulcl 21593 . . . . 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 22532 . . . 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 21659 . . . . . . 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 21587 . . . . . 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 8856 . . . . . . 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 21666 . . . . . . . 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 2318 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  C )  x.  A )  =  ( x  .ih  ( ( * `  A )  .h  C ) ) )
2524oveq1d 5873 . . . . 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 2317 . . . 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 4106 . . 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 2318 . 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 2318 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 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   CCcc 8735    x. cmul 8742   *ccj 11581   ~Hchil 21499    .h csm 21501    .ih csp 21502    ketbra ck 21537
This theorem is referenced by:  kbass6  22701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hilex 21579  ax-hfvmul 21585  ax-hvmulass 21587  ax-hfi 21658  ax-his1 21661  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-kb 22431
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