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Axiom ax-his2 21678
Description: Distributive law for inner product. Postulate (S2) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-his2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  .ih  C )  =  ( ( A 
.ih  C )  +  ( B  .ih  C
) ) )

Detailed syntax breakdown of Axiom ax-his2
StepHypRef Expression
1 cA . . . 4  class  A
2 chil 21515 . . . 4  class  ~H
31, 2wcel 1696 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1696 . . 3  wff  B  e. 
~H
6 cC . . . 4  class  C
76, 2wcel 1696 . . 3  wff  C  e. 
~H
83, 5, 7w3a 934 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H  /\  C  e. 
~H )
9 cva 21516 . . . . 5  class  +h
101, 4, 9co 5874 . . . 4  class  ( A  +h  B )
11 csp 21518 . . . 4  class  .ih
1210, 6, 11co 5874 . . 3  class  ( ( A  +h  B ) 
.ih  C )
131, 6, 11co 5874 . . . 4  class  ( A 
.ih  C )
144, 6, 11co 5874 . . . 4  class  ( B 
.ih  C )
15 caddc 8756 . . . 4  class  +
1613, 14, 15co 5874 . . 3  class  ( ( A  .ih  C )  +  ( B  .ih  C ) )
1712, 16wceq 1632 . 2  wff  ( ( A  +h  B ) 
.ih  C )  =  ( ( A  .ih  C )  +  ( B 
.ih  C ) )
188, 17wi 4 1  wff  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  .ih  C )  =  ( ( A 
.ih  C )  +  ( B  .ih  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  his7  21685  hiassdi  21686  his2sub  21687  normlem0  21704  normlem8  21712  ocsh  21878  pjspansn  22172  pjadjii  22269  braadd  22541  lnopunilem1  22606  hmops  22616  cnlnadjlem2  22664  adjadd  22689  leopadd  22728
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