Theorem hvpncan3

Description: Subtraction and addition of equal Hilbert space vectors. (Contributed by NM, 27-Aug-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	hvpncan3	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐴 ) ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hvaddsubass	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐴 ) ) )
2	1	3anidm13	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐴 ) ) )
3		hvpncan2	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 +ℎ 𝐵 ) −ℎ 𝐴 ) = 𝐵 )
4	2 3	eqtr3d	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ ( 𝐵 −ℎ 𝐴 ) ) = 𝐵 )