Theorem orthcom

Description: Orthogonality commutes. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	orthcom	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( 𝐵 ·ih 𝐴 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = ( ∗ ‘ 0 ) )
2		cj0	⊢ ( ∗ ‘ 0 ) = 0
3	1 2	eqtrdi	⊢ ( ( 𝐴 ·ih 𝐵 ) = 0 → ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = 0 )
4		ax-his1	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) )
5	4	ancoms	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐵 ·ih 𝐴 ) = ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) )
6	5	eqeq1d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐵 ·ih 𝐴 ) = 0 ↔ ( ∗ ‘ ( 𝐴 ·ih 𝐵 ) ) = 0 ) )
7	3 6	syl5ibr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 → ( 𝐵 ·ih 𝐴 ) = 0 ) )
8		fveq2	⊢ ( ( 𝐵 ·ih 𝐴 ) = 0 → ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) = ( ∗ ‘ 0 ) )
9	8 2	eqtrdi	⊢ ( ( 𝐵 ·ih 𝐴 ) = 0 → ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) = 0 )
10		ax-his1	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih 𝐵 ) = ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) )
11	10	eqeq1d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( ∗ ‘ ( 𝐵 ·ih 𝐴 ) ) = 0 ) )
12	9 11	syl5ibr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐵 ·ih 𝐴 ) = 0 → ( 𝐴 ·ih 𝐵 ) = 0 ) )
13	7 12	impbid	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih 𝐵 ) = 0 ↔ ( 𝐵 ·ih 𝐴 ) = 0 ) )