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Metamath Proof Explorer

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 ) )