Theorem ipval2lem3

Description: Lemma for ipval3 . (Contributed by NM, 1-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dipfval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	dipfval.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	dipfval.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	dipfval.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
	dipfval.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion	ipval2lem3	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		dipfval.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		dipfval.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		dipfval.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		dipfval.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
5		dipfval.7	⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
6	1 3	nvsid	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) → ( 1 𝑆 𝐵 ) = 𝐵 )
7	6	oveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) = ( 𝐴 𝐺 𝐵 ) )
8	7	fveq2d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) ) = ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) )
9	8	oveq1d	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) ) ↑ 2 ) = ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) )
10	9	3adant2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) ) ↑ 2 ) = ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) )
11		ax-1cn	⊢ 1 ∈ ℂ
12	1 2 3 4 5	ipval2lem2	⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 1 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
13	11 12	mpan2	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 1 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
14	10 13	eqeltrrd	⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) ∈ ℝ )