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Metamath Proof Explorer

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 ) ) )