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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 ∈ ℋ → ( ( 𝐴 ·_ih 𝐴 ) = 0 ↔ 𝐴 = 0_ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		ax-his4	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ ) → 0 < ( 𝐴 ·_ih 𝐴 ) )
2	1	gt0ne0d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0_ℎ ) → ( 𝐴 ·_ih 𝐴 ) ≠ 0 )
3	2	ex	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ≠ 0_ℎ → ( 𝐴 ·_ih 𝐴 ) ≠ 0 ) )
4	3	necon4d	⊢ ( 𝐴 ∈ ℋ → ( ( 𝐴 ·_ih 𝐴 ) = 0 → 𝐴 = 0_ℎ ) )
5		hi01	⊢ ( 𝐴 ∈ ℋ → ( 0_ℎ ·_ih 𝐴 ) = 0 )
6		oveq1	⊢ ( 𝐴 = 0_ℎ → ( 𝐴 ·_ih 𝐴 ) = ( 0_ℎ ·_ih 𝐴 ) )
7	6	eqeq1d	⊢ ( 𝐴 = 0_ℎ → ( ( 𝐴 ·_ih 𝐴 ) = 0 ↔ ( 0_ℎ ·_ih 𝐴 ) = 0 ) )
8	5 7	syl5ibrcom	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 = 0_ℎ → ( 𝐴 ·_ih 𝐴 ) = 0 ) )
9	4 8	impbid	⊢ ( 𝐴 ∈ ℋ → ( ( 𝐴 ·_ih 𝐴 ) = 0 ↔ 𝐴 = 0_ℎ ) )