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Axiom ax-hvdistr1 21604
Description: Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvdistr1  |-  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
) )

Detailed syntax breakdown of Axiom ax-hvdistr1
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
5 chil 21515 . . . 4  class  ~H
64, 5wcel 1696 . . 3  wff  B  e. 
~H
7 cC . . . 4  class  C
87, 5wcel 1696 . . 3  wff  C  e. 
~H
93, 6, 8w3a 934 . 2  wff  ( A  e.  CC  /\  B  e.  ~H  /\  C  e. 
~H )
10 cva 21516 . . . . 5  class  +h
114, 7, 10co 5874 . . . 4  class  ( B  +h  C )
12 csm 21517 . . . 4  class  .h
131, 11, 12co 5874 . . 3  class  ( A  .h  ( B  +h  C ) )
141, 4, 12co 5874 . . . 4  class  ( A  .h  B )
151, 7, 12co 5874 . . . 4  class  ( A  .h  C )
1614, 15, 10co 5874 . . 3  class  ( ( A  .h  B )  +h  ( A  .h  C ) )
1713, 16wceq 1632 . 2  wff  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
)
189, 17wi 4 1  wff  ( ( A  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  .h  ( B  +h  C ) )  =  ( ( A  .h  B )  +h  ( A  .h  C )
) )
Colors of variables: wff set class
This axiom is referenced by:  hvsub4  21632  hvsubass  21639  hvsubdistr1  21644  hvdistr1i  21646  hv2times  21656  hilvc  21757  hhssnv  21857  shscli  21912  spanunsni  22174  hoadddi  22399  lnopmi  22596  lnophsi  22597
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