Theorem diaelval

Description: Member of the partial isomorphism A for a lattice K . (Contributed by NM, 3-Dec-2013)

	Ref	Expression
Hypotheses	diaval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	diaval.l	⊢ ≤ = ( le ‘ 𝐾 )
	diaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	diaval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	diaval.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	diaval.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
Assertion	diaelval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		diaval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		diaval.l	⊢ ≤ = ( le ‘ 𝐾 )
3		diaval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		diaval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		diaval.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		diaval.i	⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7	1 2 3 4 5 6	diaval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } )
8	7	eleq2d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ↔ 𝐹 ∈ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ) )
9		fveq2	⊢ ( 𝑓 = 𝐹 → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝐹 ) )
10	9	breq1d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 ↔ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) )
11	10	elrab	⊢ ( 𝐹 ∈ { 𝑓 ∈ 𝑇 ∣ ( 𝑅 ‘ 𝑓 ) ≤ 𝑋 } ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) )
12	8 11	bitrdi	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) )