Theorem pjinormi

Description: The inner product of a projection and its argument is the square of the norm of the projection. Remark in Halmos p. 44. (Contributed by NM, 2-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjadjt.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjinormi	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2$$

Proof

Step	Hyp	Ref	Expression
1		pjadjt.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2		fveq2	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)={\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)$$
3		id	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)$$
4	2 3	oveq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A={\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)$$
5		2fveq3	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)={norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)$$
6	5	oveq1d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to {{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)}^2$$
7	4 6	eqeq12d	$$⊢A=if\left(A\in \mathscr{H},A,0_{ℎ}\right)\to \left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2\leftrightarrow {\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)}^2\right)$$
8		ifhvhv0	$$⊢if\left(A\in \mathscr{H},A,0_{ℎ}\right)\in \mathscr{H}$$
9	1 8	pjinormii	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right){\cdot }_{\mathrm{ih}}if\left(A\in \mathscr{H},A,0_{ℎ}\right)={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(if\left(A\in \mathscr{H},A,0_{ℎ}\right)\right)\right)}^2$$
10	7 9	dedth	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2$$