Theorem his6

Description: Zero inner product with self means vector is zero. Lemma 3.1(S6) of Beran p. 95. (Contributed by NM, 27-Jul-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	his6	|- ( A e. ~H -> ( ( A .ih A ) = 0 <-> A = 0h ) )

Proof

Step	Hyp	Ref	Expression
1		ax-his4	|- ( ( A e. ~H /\ A =/= 0h ) -> 0 < ( A .ih A ) )
2	1	gt0ne0d	|- ( ( A e. ~H /\ A =/= 0h ) -> ( A .ih A ) =/= 0 )
3	2	ex	|- ( A e. ~H -> ( A =/= 0h -> ( A .ih A ) =/= 0 ) )
4	3	necon4d	|- ( A e. ~H -> ( ( A .ih A ) = 0 -> A = 0h ) )
5		hi01	|- ( A e. ~H -> ( 0h .ih A ) = 0 )
6		oveq1	|- ( A = 0h -> ( A .ih A ) = ( 0h .ih A ) )
7	6	eqeq1d	|- ( A = 0h -> ( ( A .ih A ) = 0 <-> ( 0h .ih A ) = 0 ) )
8	5 7	syl5ibrcom	|- ( A e. ~H -> ( A = 0h -> ( A .ih A ) = 0 ) )
9	4 8	impbid	|- ( A e. ~H -> ( ( A .ih A ) = 0 <-> A = 0h ) )