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Metamath Proof Explorer

Theorem dia1eldmN

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

		Ref	Expression
	Hypotheses	dia1eldm.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
		dia1eldm.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
	Assertion	dia1eldmN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ dom 𝐼 )

Proof

Step	Hyp	Ref	Expression
1		dia1eldm.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dia1eldm.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
3		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
4	3 1	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
5	4	adantl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
6		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
8	3 7	latref	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → 𝑊 ( le ‘ 𝐾 ) 𝑊 )
9	6 4 8	syl2an	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( le ‘ 𝐾 ) 𝑊 )
10	3 7 1 2	diaeldm	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑊 ∈ dom 𝐼 ↔ ( 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ( le ‘ 𝐾 ) 𝑊 ) ) )
11	5 9 10	mpbir2and	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ dom 𝐼 )