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Theorem hiassdi 22554
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 22477 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
2 ax-his2 22546 . . . 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 1154 . . 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 458 . 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 22547 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  D  e.  ~H )  ->  (
( A  .h  B
)  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
653expa 1153 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  D  e.  ~H )  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
76adantrl 697 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B 
.ih  D ) ) )
87oveq1d 6063 . 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 2444 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 359    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952    + caddc 8957    x. cmul 8959   ~Hchil 22383    +h cva 22384    .h csm 22385    .ih csp 22386
This theorem is referenced by:  unoplin  23384  hmoplin  23406
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-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371  ax-hfvmul 22469  ax-his2 22546  ax-his3 22547
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051
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