HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ax-hvdistr1 Structured version   Unicode version

Axiom ax-hvdistr1 22511
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 8988 . . . 4  class  CC
31, 2wcel 1725 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
5 chil 22422 . . . 4  class  ~H
64, 5wcel 1725 . . 3  wff  B  e. 
~H
7 cC . . . 4  class  C
87, 5wcel 1725 . . 3  wff  C  e. 
~H
93, 6, 8w3a 936 . 2  wff  ( A  e.  CC  /\  B  e.  ~H  /\  C  e. 
~H )
10 cva 22423 . . . . 5  class  +h
114, 7, 10co 6081 . . . 4  class  ( B  +h  C )
12 csm 22424 . . . 4  class  .h
131, 11, 12co 6081 . . 3  class  ( A  .h  ( B  +h  C ) )
141, 4, 12co 6081 . . . 4  class  ( A  .h  B )
151, 7, 12co 6081 . . . 4  class  ( A  .h  C )
1614, 15, 10co 6081 . . 3  class  ( ( A  .h  B )  +h  ( A  .h  C ) )
1713, 16wceq 1652 . 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  22539  hvsubass  22546  hvsubdistr1  22551  hvdistr1i  22553  hv2times  22563  hilvc  22664  hhssnv  22764  shscli  22819  spanunsni  23081  hoadddi  23306  lnopmi  23503  lnophsi  23504
  Copyright terms: Public domain W3C validator