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

Proof

Step	Hyp	Ref	Expression
1		hvaddsubass	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\wedge A\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=A{+}_{ℎ}\left(B{-}_{ℎ}A\right)$$
2	1	3anidm13	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=A{+}_{ℎ}\left(B{-}_{ℎ}A\right)$$
3		hvpncan2	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to \left(A{+}_{ℎ}B\right){-}_{ℎ}A=B$$
4	2 3	eqtr3d	$$⊢\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\to A{+}_{ℎ}\left(B{-}_{ℎ}A\right)=B$$