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Axiom ax-hvdistr2 21605
Description: Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvdistr2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )

Detailed syntax breakdown of Axiom ax-hvdistr2
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 8751 . . . 4  class  CC
31, 2wcel 1696 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1696 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
7 chil 21515 . . . 4  class  ~H
86, 7wcel 1696 . . 3  wff  C  e. 
~H
93, 5, 8w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e. 
~H )
10 caddc 8756 . . . . 5  class  +
111, 4, 10co 5874 . . . 4  class  ( A  +  B )
12 csm 21517 . . . 4  class  .h
1311, 6, 12co 5874 . . 3  class  ( ( A  +  B )  .h  C )
141, 6, 12co 5874 . . . 4  class  ( A  .h  C )
154, 6, 12co 5874 . . . 4  class  ( B  .h  C )
16 cva 21516 . . . 4  class  +h
1714, 15, 16co 5874 . . 3  class  ( ( A  .h  C )  +h  ( B  .h  C ) )
1813, 17wceq 1632 . 2  wff  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C )
)
199, 18wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  hvsubid  21621  hvsubdistr2  21645  hv2times  21656  hilvc  21757  hhssnv  21857  hoadddir  22400  superpos  22950
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