Theorem hlipass

Description: Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlipass.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlipass.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
	hlipass.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
Assertion	hlipass	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in X\wedge C\in X\right)\right)\to \left(ASB\right)PC=A\left(BPC\right)$$

Proof

Step	Hyp	Ref	Expression
1		hlipass.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hlipass.4	$$⊢S={\cdot }_{\mathrm{𝑠OLD}}\left(U\right)$$
3		hlipass.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
4		hlph	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in {CPreHil}_{\mathrm{OLD}}$$
5	1 2 3	dipass	$$⊢\left(U\in {CPreHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in X\wedge C\in X\right)\right)\to \left(ASB\right)PC=A\left(BPC\right)$$
6	4 5	sylan	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge \left(A\in ℂ\wedge B\in X\wedge C\in X\right)\right)\to \left(ASB\right)PC=A\left(BPC\right)$$