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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 = ( +𝑣 ‘ 𝑈 )
dipfval.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
dipfval.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
dipfval.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion ipval2lem2 ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 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 = ( +𝑣 ‘ 𝑈 )
3 dipfval.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4 dipfval.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
5 dipfval.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
6 simpl1 ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → 𝑈 ∈ NrmCVec )
7 simpl2 ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → 𝐴 ∈ 𝑋 )
8 1 3 nvscl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ 𝑋 ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
9 8 3com23 ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
10 9 3expa ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
11 10 3adantl2 ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 )
12 1 2 nvgcl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐶 𝑆 𝐵 ) ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ∈ 𝑋 )
13 6 7 11 12 syl3anc ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 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 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ∈ ℝ )
16 15 resqcld ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝑁 ‘ ( 𝐴 𝐺 ( 𝐶 𝑆 𝐵 ) ) ) ↑ 2 ) ∈ ℝ )