Theorem hvdistr1i 27908

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

Step	Hyp	Ref	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 27865	. 2 |- ( ( A e. CC /\ B e. ~H /\ C e. ~H ) -> ( A .h ( B +h C ) ) = ( ( A .h B ) +h ( A .h C ) ) )
5	1, 2, 3, 4	mp3an 1424	1 |- ( A .h ( B +h C ) ) = ( ( A .h B ) +h ( A .h C ) )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   ~Hchil 27776   +h cva 27777   .h csm 27778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvdistr1 27865

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  hvsubsub4i  27916  hvnegdii  27919  pjmulii  28536  lnophmlem2  28876