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Metamath Proof Explorer

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 imbitrrid ⊢ ( ( 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 imbitrrid ⊢ ( ( 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 ) )