Theorem hlpar

Description: The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlpar.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlpar.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
	hlpar.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
	hlpar.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
Assertion	hlpar	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝐵 ) ↑ 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlpar.1	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlpar.2	⊢ 𝐺 = ( +𝑣 ‘ 𝑈 )
3		hlpar.4	⊢ 𝑆 = ( ·𝑠OLD ‘ 𝑈 )
4		hlpar.6	⊢ 𝑁 = ( normCV ‘ 𝑈 )
5		hlph	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CPreHilOLD )
6	1 2 3 4	phpar	⊢ ( ( 𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝐵 ) ↑ 2 ) ) ) )
7	5 6	syl3an1	⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( ( ( 𝑁 ‘ ( 𝐴 𝐺 𝐵 ) ) ↑ 2 ) + ( ( 𝑁 ‘ ( 𝐴 𝐺 ( - 1 𝑆 𝐵 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( 𝑁 ‘ 𝐴 ) ↑ 2 ) + ( ( 𝑁 ‘ 𝐵 ) ↑ 2 ) ) ) )