Theorem dia0eldmN

Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dia0eldm.z	⊢ 0 = ( 0. ‘ 𝐾 )
	dia0eldm.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dia0eldm.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dia0eldmN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ∈ dom 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dia0eldm.z	⊢ 0 = ( 0. ‘ 𝐾 )
2		dia0eldm.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dia0eldm.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4		hlop	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
5	4	adantr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ OP )
6		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7	6 1	op0cl	⊢ ( 𝐾 ∈ OP → 0 ∈ ( Base ‘ 𝐾 ) )
8	5 7	syl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ∈ ( Base ‘ 𝐾 ) )
9	6 2	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
10		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
11	6 10 1	op0le	⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 0 ( le ‘ 𝐾 ) 𝑊 )
12	4 9 11	syl2an	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ( le ‘ 𝐾 ) 𝑊 )
13	6 10 2 3	diaeldm	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 0 ∈ dom 𝐼 ↔ ( 0 ∈ ( Base ‘ 𝐾 ) ∧ 0 ( le ‘ 𝐾 ) 𝑊 ) ) )
14	8 12 13	mpbir2and	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 0 ∈ dom 𝐼 )