Theorem dibelval1st2N

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dibelval1st2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dibelval1st2.l	⊢ ≤ = ( le ‘ 𝐾 )
	dibelval1st2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibelval1st2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dibelval1st2.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	dibelval1st2.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibelval1st2N	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		dibelval1st2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dibelval1st2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dibelval1st2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dibelval1st2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dibelval1st2.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		dibelval1st2.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
8	1 2 3 7 6	dibelval1st	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
9	1 2 3 4 5 7	diatrl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )
10	8 9	syld3an3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 𝑅 ‘ ( 1st ‘ 𝑌 ) ) ≤ 𝑋 )