Theorem dvh1dimat

Description: There exists an atom. (Contributed by NM, 25-Apr-2015)

	Ref	Expression
Hypotheses	dvh4dimat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dvh4dimat.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dvh1dimat.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	dvh1dimat.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	dvh1dimat	⊢ ( 𝜑 → ∃ 𝑠 𝑠 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dvh4dimat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvh4dimat.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		dvh1dimat.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
4		dvh1dimat.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5		eqid	⊢ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
7	1 5 6 2 3 4	dihat	⊢ ( 𝜑 → ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ 𝐴 )
8		elex2	⊢ ( ( ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ 𝐴 → ∃ 𝑠 𝑠 ∈ 𝐴 )
9	7 8	syl	⊢ ( 𝜑 → ∃ 𝑠 𝑠 ∈ 𝐴 )