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Theorem hiassdi 21670
Description: Distributive/associative law for inner product, useful for linearity proofs. (Contributed by NM, 10-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hiassdi  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  .h  B
)  +h  C ) 
.ih  D )  =  ( ( A  x.  ( B  .ih  D ) )  +  ( C 
.ih  D ) ) )

Proof of Theorem hiassdi
StepHypRef Expression
1 hvmulcl 21593 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
2 ax-his2 21662 . . . 4  |-  ( ( ( A  .h  B
)  e.  ~H  /\  C  e.  ~H  /\  D  e.  ~H )  ->  (
( ( A  .h  B )  +h  C
)  .ih  D )  =  ( ( ( A  .h  B ) 
.ih  D )  +  ( C  .ih  D
) ) )
323expb 1152 . . 3  |-  ( ( ( A  .h  B
)  e.  ~H  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  .h  B
)  +h  C ) 
.ih  D )  =  ( ( ( A  .h  B )  .ih  D )  +  ( C 
.ih  D ) ) )
41, 3sylan 457 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  .h  B
)  +h  C ) 
.ih  D )  =  ( ( ( A  .h  B )  .ih  D )  +  ( C 
.ih  D ) ) )
5 ax-his3 21663 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  D  e.  ~H )  ->  (
( A  .h  B
)  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
653expa 1151 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  D  e.  ~H )  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
76adantrl 696 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B 
.ih  D ) ) )
87oveq1d 5873 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  .h  B
)  .ih  D )  +  ( C  .ih  D ) )  =  ( ( A  x.  ( B  .ih  D ) )  +  ( C  .ih  D ) ) )
94, 8eqtrd 2315 1  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( (
( A  .h  B
)  +h  C ) 
.ih  D )  =  ( ( A  x.  ( B  .ih  D ) )  +  ( C 
.ih  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502
This theorem is referenced by:  unoplin  22500  hmoplin  22522
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hfvmul 21585  ax-his2 21662  ax-his3 21663
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861
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