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Metamath Proof Explorer

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 ) ) )