Theorem elpjch

Description: Reconstruction of the subspace of a projection operator. Part of Theorem 26.2 of Halmos p. 44. (Contributed by NM, 24-Apr-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	elpjch	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\to \left(\mathrm{ran}T\in {\mathbf{C}}_{\mathscr{H}}\wedge T={\mathrm{proj}}_{ℎ}\left(\mathrm{ran}T\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dfpjop	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\leftrightarrow \left(T\in \mathrm{HrmOp}\wedge T\circ T=T\right)$$
2		hmopidmch	$$⊢\left(T\in \mathrm{HrmOp}\wedge T\circ T=T\right)\to \mathrm{ran}T\in {\mathbf{C}}_{\mathscr{H}}$$
3		hmopidmpj	$$⊢\left(T\in \mathrm{HrmOp}\wedge T\circ T=T\right)\to T={\mathrm{proj}}_{ℎ}\left(\mathrm{ran}T\right)$$
4	2 3	jca	$$⊢\left(T\in \mathrm{HrmOp}\wedge T\circ T=T\right)\to \left(\mathrm{ran}T\in {\mathbf{C}}_{\mathscr{H}}\wedge T={\mathrm{proj}}_{ℎ}\left(\mathrm{ran}T\right)\right)$$
5	1 4	sylbi	$$⊢T\in \mathrm{ran}{\mathrm{proj}}_{ℎ}\to \left(\mathrm{ran}T\in {\mathbf{C}}_{\mathscr{H}}\wedge T={\mathrm{proj}}_{ℎ}\left(\mathrm{ran}T\right)\right)$$