Theorem hlhils0

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

	Ref	Expression
Hypotheses	hlhilsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	hlhilsbase.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐿 )
	hlhilsbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
	hlhilsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
	hlhilsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	hlhils0.z	⊢ 0 = ( 0g ‘ 𝑆 )
Assertion	hlhils0	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		hlhilsbase.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		hlhilsbase.l	⊢ 𝐿 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		hlhilsbase.s	⊢ 𝑆 = ( Scalar ‘ 𝐿 )
4		hlhilsbase.u	⊢ 𝑈 = ( ( HLHil ‘ 𝐾 ) ‘ 𝑊 )
5		hlhilsbase.r	⊢ 𝑅 = ( Scalar ‘ 𝑈 )
6		hlhilsbase.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		hlhils0.z	⊢ 0 = ( 0g ‘ 𝑆 )
8		eqidd	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
9		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
10	1 2 3 4 5 6 9	hlhilsbase2	⊢ ( 𝜑 → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑅 ) )
11		eqid	⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
12	1 2 3 4 5 6 11	hlhilsplus2	⊢ ( 𝜑 → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑅 ) )
13	12	oveqdr	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) )
14	8 10 13	grpidpropd	⊢ ( 𝜑 → ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑅 ) )
15	7 14	syl5eq	⊢ ( 𝜑 → 0 = ( 0g ‘ 𝑅 ) )