Theorem pjige0i

Description: The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	pjadjt.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
Assertion	pjige0i	$$⊢A\in \mathscr{H}\to 0\le {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A$$

Proof

Step	Hyp	Ref	Expression
1		pjadjt.1	$$⊢H\in {\mathbf{C}}_{\mathscr{H}}$$
2	1	pjhcli	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}$$
3		normcl	$$⊢{\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\in \mathscr{H}\to {norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)\in ℝ$$
4	2 3	syl	$$⊢A\in \mathscr{H}\to {norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)\in ℝ$$
5	4	sqge0d	$$⊢A\in \mathscr{H}\to 0\le {{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2$$
6	1	pjinormi	$$⊢A\in \mathscr{H}\to {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A={{norm}_{ℎ}\left({\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right)\right)}^2$$
7	5 6	breqtrrd	$$⊢A\in \mathscr{H}\to 0\le {\mathrm{proj}}_{ℎ}\left(H\right)\left(A\right){\cdot }_{\mathrm{ih}}A$$