Theorem dihopelvalbN

Description: Ordered pair member of the partial isomorphism H for argument under W . (Contributed by NM, 21-Mar-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dihval3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	dihval3.l	⊢ ≤ = ( le ‘ 𝐾 )
	dihval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dihval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	dihval3.o	⊢ 𝑂 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
	dihval3.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
Assertion	dihopelvalbN	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 𝑂 ) ) )

Proof

Step	Hyp	Ref	Expression
1		dihval3.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihval3.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dihval3.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dihval3.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dihval3.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6		dihval3.o	⊢ 𝑂 = ( 𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
7		dihval3.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
8		eqid	⊢ ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
9	1 2 3 7 8	dihvalb	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) )
10	9	eleq2d	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ) )
11	1 2 3 4 5 6 8	dibopelval3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 𝑂 ) ) )
12	10 11	bitrd	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑋 ) ↔ ( ( 𝐹 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝐹 ) ≤ 𝑋 ) ∧ 𝑆 = 𝑂 ) ) )