Theorem dihat

Description: There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014)

	Ref	Expression
Hypotheses	dihat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihat.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
	dihat.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihat.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihat.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
	dihat.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion	dihat	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		dihat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dihat.p	⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
3		dihat.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4		dihat.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dihat.a	⊢ 𝐴 = ( LSAtoms ‘ 𝑈 )
6		dihat.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		eqid	⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
8		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
9	7 8 1	lhpocat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) )
10	6 9	syl	⊢ ( 𝜑 → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Atoms ‘ 𝐾 ) )
11	2 10	eqeltrid	⊢ ( 𝜑 → 𝑃 ∈ ( Atoms ‘ 𝐾 ) )
12	8 1 4 3 5	dihatlat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ ( Atoms ‘ 𝐾 ) ) → ( 𝐼 ‘ 𝑃 ) ∈ 𝐴 )
13	6 11 12	syl2anc	⊢ ( 𝜑 → ( 𝐼 ‘ 𝑃 ) ∈ 𝐴 )