HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ax-hvdistr2 Unicode version

Axiom ax-hvdistr2 21419
Description: Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-hvdistr2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )

Detailed syntax breakdown of Axiom ax-hvdistr2
StepHypRef Expression
1 cA . . . 4  class  A
2 cc 8615 . . . 4  class  CC
31, 2wcel 1621 . . 3  wff  A  e.  CC
4 cB . . . 4  class  B
54, 2wcel 1621 . . 3  wff  B  e.  CC
6 cC . . . 4  class  C
7 chil 21329 . . . 4  class  ~H
86, 7wcel 1621 . . 3  wff  C  e. 
~H
93, 5, 8w3a 939 . 2  wff  ( A  e.  CC  /\  B  e.  CC  /\  C  e. 
~H )
10 caddc 8620 . . . . 5  class  +
111, 4, 10co 5710 . . . 4  class  ( A  +  B )
12 csm 21331 . . . 4  class  .h
1311, 6, 12co 5710 . . 3  class  ( ( A  +  B )  .h  C )
141, 6, 12co 5710 . . . 4  class  ( A  .h  C )
154, 6, 12co 5710 . . . 4  class  ( B  .h  C )
16 cva 21330 . . . 4  class  +h
1714, 15, 16co 5710 . . 3  class  ( ( A  .h  C )  +h  ( B  .h  C ) )
1813, 17wceq 1619 . 2  wff  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C )
)
199, 18wi 6 1  wff  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )
Colors of variables: wff set class
This axiom is referenced by:  hvsubid  21435  hvsubdistr2  21459  hv2times  21470  hilvc  21571  hhssnv  21671  hoadddir  22214  superpos  22764
  Copyright terms: Public domain W3C validator