Theorem diadmleN

Description: A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	diadmle.l	⊢ ≤ = ( le ‘ 𝐾 )
	diadmle.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	diadmle.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	diadmleN	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ≤ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		diadmle.l	⊢ ≤ = ( le ‘ 𝐾 )
2		diadmle.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		diadmle.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5	4 1 2 3	diaeldm	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → ( 𝑋 ∈ dom 𝐼 ↔ ( 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑋 ≤ 𝑊 ) ) )
6	5	simplbda	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ≤ 𝑊 )