Theorem dibopelval3

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014)

	Ref	Expression
Hypotheses	dibval3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dibval3.l	⊢ ≤ = ( le ‘ 𝐾 )
	dibval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dibval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dibval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	dibval3.o	⊢ 0 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
	dibval3.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dibopelval3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 0 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dibval3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dibval3.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dibval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dibval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dibval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		dibval3.o	⊢ 0 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
7		dibval3.i	⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8		eqid	⊢ ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
9	1 2 3 4 6 8 7	dibopelval2	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ) )
10	1 2 3 4 5 8	diaelval	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ↔ ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ) )
11	10	anbi1d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ( 𝐹 ∈ ( ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ∧ 𝑆 = 0 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 0 ) ) )
12	9 11	bitrd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 0 ) ) )