Theorem hlass

Description: Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hladdf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hladdf.2	$$⊢G={+}_v\left(U\right)$$
Assertion	hlass	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=AG\left(BGC\right)$$

Proof

Step	Hyp	Ref	Expression
1		hladdf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hladdf.2	$$⊢G={+}_v\left(U\right)$$
3		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
4	1 2	nvass	$$⊢\left(U\in \mathrm{NrmCVec}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=AG\left(BGC\right)$$
5	3 4	sylan	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGB\right)GC=AG\left(BGC\right)$$