Metamath Proof Explorer


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 = ( +𝑣𝑈 )
dipfval.4 𝑆 = ( ·𝑠OLD𝑈 )
dipfval.6 𝑁 = ( normCV𝑈 )
dipfval.7 𝑃 = ( ·𝑖OLD𝑈 )
Assertion ipval2lem4 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 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 = ( +𝑣𝑈 )
3 dipfval.4 𝑆 = ( ·𝑠OLD𝑈 )
4 dipfval.6 𝑁 = ( normCV𝑈 )
5 dipfval.7 𝑃 = ( ·𝑖OLD𝑈 )
6 1 2 3 4 5 ipval2lem2 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )
7 6 recnd ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℂ )