Theorem hlipcj

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

	Ref	Expression
Hypotheses	hlipf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	hlipf.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
Assertion	hlipcj	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\wedge B\in X\right)\to APB=\overline{BPA}$$

Proof

Step	Hyp	Ref	Expression
1		hlipf.1	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		hlipf.7	$$⊢P={\cdot }_{\mathrm{𝑖OLD}}\left(U\right)$$
3		hlnv	$$⊢U\in {CHil}_{\mathrm{OLD}}\to U\in \mathrm{NrmCVec}$$
4	1 2	dipcj	$$⊢\left(U\in \mathrm{NrmCVec}\wedge B\in X\wedge A\in X\right)\to \overline{BPA}=APB$$
5	3 4	syl3an1	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge B\in X\wedge A\in X\right)\to \overline{BPA}=APB$$
6	5	3com23	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\wedge B\in X\right)\to \overline{BPA}=APB$$
7	6	eqcomd	$$⊢\left(U\in {CHil}_{\mathrm{OLD}}\wedge A\in X\wedge B\in X\right)\to APB=\overline{BPA}$$