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Theorem hiassdi 22598
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 22521 . . 3  |-  ( ( A  e.  CC  /\  B  e.  ~H )  ->  ( A  .h  B
)  e.  ~H )
2 ax-his2 22590 . . . 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 1155 . . 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 459 . 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 22591 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  D  e.  ~H )  ->  (
( A  .h  B
)  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
653expa 1154 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  D  e.  ~H )  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B  .ih  D ) ) )
76adantrl 698 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  ~H )  /\  ( C  e.  ~H  /\  D  e.  ~H )
)  ->  ( ( A  .h  B )  .ih  D )  =  ( A  x.  ( B 
.ih  D ) ) )
87oveq1d 6099 . 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 2470 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 360    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993    + caddc 8998    x. cmul 9000   ~Hchil 22427    +h cva 22428    .h csm 22429    .ih csp 22430
This theorem is referenced by:  unoplin  23428  hmoplin  23450
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-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-hfvmul 22513  ax-his2 22590  ax-his3 22591
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  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-ral 2712  df-rex 2713  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-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-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-ov 6087
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