Theorem hlhil0

Description: The zero vector for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015) (Revised by Mario Carneiro, 29-Jun-2015)

	Ref	Expression
Hypotheses	hlhil0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hlhil0.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hlhil0.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhil0.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhil0.z	⊢ 0 = ( 0g ‘ 𝐿 )
Assertion	hlhil0	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		hlhil0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhil0.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hlhil0.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
4		hlhil0.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5		hlhil0.z	⊢ 0 = ( 0g ‘ 𝐿 )
6		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) )
7		eqid	⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
8	1 3 4 2 7	hlhilbase	⊢ ( 𝜑 → ( Base ‘ 𝐿 ) = ( Base ‘ 𝑈 ) )
9		eqid	⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 )
10	1 3 4 2 9	hlhilplus	⊢ ( 𝜑 → ( +g ‘ 𝐿 ) = ( +g ‘ 𝑈 ) )
11	10	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝐿 ) ∧ 𝑦 ∈ ( Base ‘ 𝐿 ) ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑈 ) 𝑦 ) )
12	6 8 11	grpidpropd	⊢ ( 𝜑 → ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝑈 ) )
13	5 12	eqtrid	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑈 ) )