Metamath Proof Explorer


Theorem ipval2lem2

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 ipval2lem2 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )

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 simpl1 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → 𝑈 ∈ NrmCVec )
7 simpl2 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → 𝐴𝑋 )
8 1 3 nvscl ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ ℂ ∧ 𝐵𝑋 ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
9 8 3com23 ( ( 𝑈 ∈ NrmCVec ∧ 𝐵𝑋𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
10 9 3expa ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
11 10 3adantl2 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
12 1 2 nvgcl ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋 ∧ ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ∈ 𝑋 )
13 6 7 11 12 syl3anc ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ∈ 𝑋 )
14 1 4 nvcl ( ( 𝑈 ∈ NrmCVec ∧ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ∈ ℝ )
15 6 13 14 syl2anc ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ∈ ℝ )
16 15 resqcld ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴𝑋𝐵𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )