Theorem hvmulcom 27900

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

Step	Hyp	Ref	Expression
1		mulcom 10022	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2	1	oveq1d 6665	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) .h C ) = ( ( B x. A ) .h C ) )
3	2	3adant3 1081	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( ( B x. A ) .h C ) )
4		ax-hvmulass 27864	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( A x. B ) .h C ) = ( A .h ( B .h C ) ) )
5		ax-hvmulass 27864	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. ~H ) -> ( ( B x. A ) .h C ) = ( B .h ( A .h C ) ) )
6	5	3com12 1269	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( ( B x. A ) .h C ) = ( B .h ( A .h C ) ) )
7	3, 4, 6	3eqtr3d 2664	1 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( A .h ( B .h C ) ) = ( B .h ( A .h C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   x. cmul 9941   ~Hchil 27776   .h csm 27778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-mulcom 10000  ax-hvmulass 27864

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  hvmulcomi  27904  hvsubdistr1  27906  lnopmi  28859