Theorem hvadd4i

Description: Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hvass.1	$$⊢A\in \mathscr{H}$$
	hvass.2	$$⊢B\in \mathscr{H}$$
	hvass.3	$$⊢C\in \mathscr{H}$$
	hvadd4.4	$$⊢D\in \mathscr{H}$$
Assertion	hvadd4i	$$⊢\left(A{+}_{ℎ}B\right){+}_{ℎ}\left(C{+}_{ℎ}D\right)=\left(A{+}_{ℎ}C\right){+}_{ℎ}\left(B{+}_{ℎ}D\right)$$

Proof

Step	Hyp	Ref	Expression
1		hvass.1	$$⊢A\in \mathscr{H}$$
2		hvass.2	$$⊢B\in \mathscr{H}$$
3		hvass.3	$$⊢C\in \mathscr{H}$$
4		hvadd4.4	$$⊢D\in \mathscr{H}$$
5		hvadd4	$$⊢\left(\left(A\in \mathscr{H}\wedge B\in \mathscr{H}\right)\wedge \left(C\in \mathscr{H}\wedge D\in \mathscr{H}\right)\right)\to \left(A{+}_{ℎ}B\right){+}_{ℎ}\left(C{+}_{ℎ}D\right)=\left(A{+}_{ℎ}C\right){+}_{ℎ}\left(B{+}_{ℎ}D\right)$$
6	1 2 3 4 5	mp4an	$$⊢\left(A{+}_{ℎ}B\right){+}_{ℎ}\left(C{+}_{ℎ}D\right)=\left(A{+}_{ℎ}C\right){+}_{ℎ}\left(B{+}_{ℎ}D\right)$$