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	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=B$$

Proof

Step	Hyp	Ref	Expression
1		ax-hvcom	$$⊢\left(B\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to B{+}_{ℎ}A=A{+}_{ℎ}B$$
2	1	oveq1d	$$⊢\left(B\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to \left(B{+}_{ℎ}A\right){-}_{ℎ}A=\left(A{+}_{ℎ}B\right){-}_{ℎ}A$$
3		hvpncan	$$⊢\left(B\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to \left(B{+}_{ℎ}A\right){-}_{ℎ}A=B$$
4	2 3	eqtr3d	$$⊢\left(B\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=B$$
5	4	ancoms	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=B$$