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Metamath Proof Explorer

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 )