Metamath Proof Explorer

Theorem ip1i

Description: Equation 6.47 of Ponnusamy p. 362. (Contributed by NM, 27-Apr-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ip1i.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
ip1i.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
ip1i.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
ip1i.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
ip1i.9 ⊢ 𝑈 ∈ CPreHilOLD
ip1i.a ⊢ 𝐴 ∈ 𝑋
ip1i.b ⊢ 𝐵 ∈ 𝑋
ip1i.c ⊢ 𝐶 ∈ 𝑋
Assertion ip1i ( ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) + ( ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) 𝑃 𝐶 ) ) = ( 2 · ( 𝐴 𝑃 𝐶 ) )

Proof

Step Hyp Ref Expression
1 ip1i.1 ⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2 ip1i.2 ⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3 ip1i.4 ⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4 ip1i.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
5 ip1i.9 ⊢ 𝑈 ∈ CPreHilOLD
6 ip1i.a ⊢ 𝐴 ∈ 𝑋
7 ip1i.b ⊢ 𝐵 ∈ 𝑋
8 ip1i.c ⊢ 𝐶 ∈ 𝑋
9 eqid ⊢ ( normCV ‘ 𝑈 ) = ( normCV ‘ 𝑈 )
10 ax-1cn ⊢ 1 ∈ ℂ
11 1 2 3 4 5 6 7 8 9 10 ip1ilem ⊢ ( ( ( 𝐴 𝐺 𝐵 ) 𝑃 𝐶 ) + ( ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) 𝑃 𝐶 ) ) = ( 2 · ( 𝐴 𝑃 𝐶 ) )