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

Theorem hvmulcom 24464
Description: Scalar multiplication commutative law. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)
Assertion
Ref Expression
hvmulcom  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )

Proof of Theorem hvmulcom
StepHypRef Expression
1 mulcom 9387 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6125 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .h  C
)  =  ( ( B  x.  A )  .h  C ) )
323adant3 1008 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( ( B  x.  A )  .h  C ) )
4 ax-hvmulass 24428 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
5 ax-hvmulass 24428 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
653com12 1191 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
73, 4, 63eqtr3d 2483 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6110   CCcc 9299    x. cmul 9306   ~Hchil 24340    .h csm 24342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-mulcom 9365  ax-hvmulass 24428
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-iota 5400  df-fv 5445  df-ov 6113
This theorem is referenced by:  hvmulcomi  24468  hvsubdistr1  24470  lnopmi  25423
  Copyright terms: Public domain W3C validator