Theorem normsqi

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

	Ref	Expression
Hypothesis	normcl.1	⊢ 𝐴 ∈ ℋ
Assertion	normsqi	⊢ ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		normcl.1	⊢ 𝐴 ∈ ℋ
2		normval	⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) )
3	1 2	ax-mp	⊢ ( normℎ ‘ 𝐴 ) = ( √ ‘ ( 𝐴 ·ih 𝐴 ) )
4	3	oveq1i	⊢ ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) ↑ 2 )
5		hiidge0	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( 𝐴 ·ih 𝐴 ) )
6	1 5	ax-mp	⊢ 0 ≤ ( 𝐴 ·ih 𝐴 )
7		hiidrcl	⊢ ( 𝐴 ∈ ℋ → ( 𝐴 ·ih 𝐴 ) ∈ ℝ )
8	1 7	ax-mp	⊢ ( 𝐴 ·ih 𝐴 ) ∈ ℝ
9	8	sqsqrti	⊢ ( 0 ≤ ( 𝐴 ·ih 𝐴 ) → ( ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) )
10	6 9	ax-mp	⊢ ( ( √ ‘ ( 𝐴 ·ih 𝐴 ) ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 )
11	4 10	eqtri	⊢ ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 )