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Metamath Proof Explorer

Theorem imsdval2

Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of Kreyszig p. 59. (Contributed by NM, 28-Nov-2006) (New usage is discouraged.)

Ref Expression
Hypotheses imsdval2.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
imsdval2.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
imsdval2.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
imsdval2.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
imsdval2.8 ⊢ 𝐷 = ( IndMet ‘ 𝑈 )
Assertion imsdval2 ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 imsdval2.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 imsdval2.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3 imsdval2.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4 imsdval2.6 ⊢ 𝑁 = ( normCV ‘ 𝑈 )
5 imsdval2.8 ⊢ 𝐷 = ( IndMet ‘ 𝑈 )
6 eqid ⊢ ( −𝑣 ‘ 𝑈 ) = ( −𝑣 ‘ 𝑈 )
7 1 6 4 5 imsdval ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) ) )
8 1 2 3 6 nvmval ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) = ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) )
9 8 fveq2d ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝐴 ( −𝑣 ‘ 𝑈 ) 𝐵 ) ) = ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) )
10 7 9 eqtrd ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) )