Theorem dibelval2nd

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

	Ref	Expression
Hypotheses	dibelval2nd.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dibelval2nd.l	⊢ ≤ = ( le ‘ 𝐾 )
	dibelval2nd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibelval2nd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dibelval2nd.o	⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
	dibelval2nd.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibelval2nd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 2nd ‘ 𝑌 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		dibelval2nd.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dibelval2nd.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dibelval2nd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dibelval2nd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dibelval2nd.o	⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6		dibelval2nd.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
7		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
8	1 2 3 4 5 7 6	dibval2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) )
9	8	eleq2d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) ) )
10	9	biimp3a	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) )
11		xp2nd	⊢ ( 𝑌 ∈ ( ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) × { 0 } ) → ( 2nd ‘ 𝑌 ) ∈ { 0 } )
12		elsni	⊢ ( ( 2nd ‘ 𝑌 ) ∈ { 0 } → ( 2nd ‘ 𝑌 ) = 0 )
13	10 11 12	3syl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ∧ 𝑌 ∈ ( 𝐼 ‘ 𝑋 ) ) → ( 2nd ‘ 𝑌 ) = 0 )