Theorem hvnegdi

Description: Distribution of negative over subtraction. (Contributed by NM, 2-Apr-2000) (New usage is discouraged.)

	Ref	Expression
Assertion	hvnegdi	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 −ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) )
2	1	oveq2d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) )
3		oveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐵 −ℎ 𝐴 ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
4	2 3	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐴 ) ↔ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
5		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
6	5	oveq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
7		oveq1	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
8	6 7	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ 𝐵 ) ) = ( 𝐵 −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↔ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
9		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
10		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
11	9 10	hvnegdii	⊢ ( - 1 ·ℎ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) −ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) −ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
12	4 8 11	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐴 ) )