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Metamath Proof Explorer

Theorem normsub

Description: Swapping order of subtraction doesn't change the norm of a vector. (Contributed by NM, 14-Aug-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	normsub	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( norm_ℎ ‘ ( 𝐴 −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		fvoveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( norm_ℎ ‘ ( 𝐴 −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ 𝐵 ) ) )
2		oveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( 𝐵 −_ℎ 𝐴 ) = ( 𝐵 −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) )
3	2	fveq2d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( norm_ℎ ‘ ( 𝐵 −_ℎ 𝐴 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) )
4	1 3	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( norm_ℎ ‘ ( 𝐴 −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ 𝐴 ) ) ↔ ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) ) )
5		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) )
6	5	fveq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) ) )
7		fvoveq1	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( norm_ℎ ‘ ( 𝐵 −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) = ( norm_ℎ ‘ ( if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) )
8	6 7	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) ↔ ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) ) = ( norm_ℎ ‘ ( if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ) ) )
9		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ ℋ
10		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ∈ ℋ
11	9 10	normsubi	⊢ ( norm_ℎ ‘ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) −_ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) ) = ( norm_ℎ ‘ ( if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) −_ℎ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) )
12	4 8 11	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( norm_ℎ ‘ ( 𝐴 −_ℎ 𝐵 ) ) = ( norm_ℎ ‘ ( 𝐵 −_ℎ 𝐴 ) ) )