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Theorem hvmulcom 21622
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 8823 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 5873 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  .h  C
)  =  ( ( B  x.  A )  .h  C ) )
323adant3 975 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( ( B  x.  A )  .h  C ) )
4 ax-hvmulass 21587 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  x.  B
)  .h  C )  =  ( A  .h  ( B  .h  C
) ) )
5 ax-hvmulass 21587 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
653com12 1155 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( B  x.  A
)  .h  C )  =  ( B  .h  ( A  .h  C
) ) )
73, 4, 63eqtr3d 2323 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  ( A  .h  ( B  .h  C ) )  =  ( B  .h  ( A  .h  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742   ~Hchil 21499    .h csm 21501
This theorem is referenced by:  hvmulcomi  21626  hvsubdistr1  21628  lnopmi  22580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-mulcom 8801  ax-hvmulass 21587
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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