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Metamath Proof Explorer

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 ) ) )