Theorem ipval2lem4

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	ipval2lem4	⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℂ )

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 2 3 4 5	ipval2lem2	⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
7	6	recnd	⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℂ )