Theorem normsqi

Description: The square of a norm. (Contributed by NM, 21-Aug-1999) (New usage is discouraged.)

	Ref	Expression
Hypothesis	normcl.1	$$⊢A\in \mathscr{H}$$
Assertion	normsqi	$$⊢{{norm}_{ℎ}\left(A\right)}^2=A{\cdot }_{\mathrm{ih}}A$$

Proof

Step	Hyp	Ref	Expression
1		normcl.1	$$⊢A\in \mathscr{H}$$
2		normval	$$⊢A\in \mathscr{H}\to {norm}_{ℎ}\left(A\right)=\sqrt{A{\cdot }_{\mathrm{ih}}A}$$
3	1 2	ax-mp	$$⊢{norm}_{ℎ}\left(A\right)=\sqrt{A{\cdot }_{\mathrm{ih}}A}$$
4	3	oveq1i	$$⊢{{norm}_{ℎ}\left(A\right)}^2={\sqrt{A{\cdot }_{\mathrm{ih}}A}}^2$$
5		hiidge0	$$⊢A\in \mathscr{H}\to 0\le A{\cdot }_{\mathrm{ih}}A$$
6	1 5	ax-mp	$$⊢0\le A{\cdot }_{\mathrm{ih}}A$$
7		hiidrcl	$$⊢A\in \mathscr{H}\to A{\cdot }_{\mathrm{ih}}A\in ℝ$$
8	1 7	ax-mp	$$⊢A{\cdot }_{\mathrm{ih}}A\in ℝ$$
9	8	sqsqrti	$$⊢0\le A{\cdot }_{\mathrm{ih}}A\to {\sqrt{A{\cdot }_{\mathrm{ih}}A}}^2=A{\cdot }_{\mathrm{ih}}A$$
10	6 9	ax-mp	$$⊢{\sqrt{A{\cdot }_{\mathrm{ih}}A}}^2=A{\cdot }_{\mathrm{ih}}A$$
11	4 10	eqtri	$$⊢{{norm}_{ℎ}\left(A\right)}^2=A{\cdot }_{\mathrm{ih}}A$$