Theorem hvpncan2

Description: Addition/subtraction cancellation law for vectors in Hilbert space. (Contributed by NM, 7-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	hvpncan2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐵 +ℎ 𝐴 ) = ( 𝐴 +ℎ 𝐵 ) )
2	1	oveq1d	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐴 ) −ℎ 𝐴 ) = ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) )
3		hvpncan	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐵 +ℎ 𝐴 ) −ℎ 𝐴 ) = 𝐵 )
4	2 3	eqtr3d	⊢ ( ( 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = 𝐵 )
5	4	ancoms	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = 𝐵 )