Theorem dibelval1st1

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

	Ref	Expression
Hypotheses	dibelval1st1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dibelval1st1.l	⊢ ≤ = ( le ‘ 𝐾 )
	dibelval1st1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibelval1st1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dibelval1st1.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibelval1st1	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		dibelval1st1.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dibelval1st1.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dibelval1st1.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dibelval1st1.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dibelval1st1.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
6		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 6 5	dibelval1st	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
8	1 2 3 4 6	diael	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ ( 1st ‘ 𝑌 ) ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )
9	7 8	syld3an3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 1st ‘ 𝑌 ) ∈ 𝑇 )