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Axiom ax-his2 21656
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 21493 . . . 4  class  ~H
31, 2wcel 1687 . . 3  wff  A  e. 
~H
4 cB . . . 4  class  B
54, 2wcel 1687 . . 3  wff  B  e. 
~H
6 cC . . . 4  class  C
76, 2wcel 1687 . . 3  wff  C  e. 
~H
83, 5, 7w3a 936 . 2  wff  ( A  e.  ~H  /\  B  e.  ~H  /\  C  e. 
~H )
9 cva 21494 . . . . 5  class  +h
101, 4, 9co 5821 . . . 4  class  ( A  +h  B )
11 csp 21496 . . . 4  class  .ih
1210, 6, 11co 5821 . . 3  class  ( ( A  +h  B ) 
.ih  C )
131, 6, 11co 5821 . . . 4  class  ( A 
.ih  C )
144, 6, 11co 5821 . . . 4  class  ( B 
.ih  C )
15 caddc 8737 . . . 4  class  +
1613, 14, 15co 5821 . . 3  class  ( ( A  .ih  C )  +  ( B  .ih  C ) )
1712, 16wceq 1625 . 2  wff  ( ( A  +h  B ) 
.ih  C )  =  ( ( A  .ih  C )  +  ( B 
.ih  C ) )
188, 17wi 6 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  21663  hiassdi  21664  his2sub  21665  normlem0  21682  normlem8  21690  ocsh  21856  pjspansn  22150  pjadjii  22247  braadd  22519  lnopunilem1  22584  hmops  22594  cnlnadjlem2  22642  adjadd  22667  leopadd  22706
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