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Theorem hvdistr1i 22506
Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvdistr1.1  |-  A  e.  CC
hvdistr1.2  |-  B  e. 
~H
hvdistr1.3  |-  C  e. 
~H
Assertion
Ref Expression
hvdistr1i  |-  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
)

Proof of Theorem hvdistr1i
StepHypRef Expression
1 hvdistr1.1 . 2  |-  A  e.  CC
2 hvdistr1.2 . 2  |-  B  e. 
~H
3 hvdistr1.3 . 2  |-  C  e. 
~H
4 ax-hvdistr1 22464 . 2  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
) )
51, 2, 3, 4mp3an 1279 1  |-  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944   ~Hchil 22375    +h cva 22376    .h csm 22377
This theorem is referenced by:  hvsubsub4i  22514  hvnegdii  22517  pjmulii  23132  lnophmlem2  23473
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-hvdistr1 22464
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938
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