Theorem hvmulcomi 27904

Description: Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hvmulcom.1	|- A e. CC
hvmulcom.2	|- B e. CC
hvmulcom.3	|- C e. ~H

Assertion

Ref	Expression
hvmulcomi	|- ( A .h ( B .h C ) ) = ( B .h ( A .h C ) )

Proof of Theorem hvmulcomi

Step	Hyp	Ref	Expression
1		hvmulcom.1	. 2 |- A e. CC
2		hvmulcom.2	. 2 |- B e. CC
3		hvmulcom.3	. 2 |- C e. ~H
4		hvmulcom 27900	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. ~H ) -> ( A .h ( B .h C ) ) = ( B .h ( A .h C ) ) )
5	1, 2, 3, 4	mp3an 1424	1 |- ( A .h ( B .h C ) ) = ( B .h ( A .h C ) )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   ~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: (None)