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Metamath Proof Explorer

Theorem dicopelval

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 15-Feb-2014)

Ref Expression
Hypotheses dicval.l ⊢ ≤ = ( le ‘ 𝐾 )
dicval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dicval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dicval.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dicval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dicval.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dicval.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dicelval.f ⊢ 𝐹 ∈ V
dicelval.s ⊢ 𝑆 ∈ V
Assertion dicopelval ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) )

Proof

Step Hyp Ref Expression
1 dicval.l ⊢ ≤ = ( le ‘ 𝐾 )
2 dicval.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 dicval.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 dicval.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
5 dicval.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 dicval.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
7 dicval.i ⊢ 𝐼 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
8 dicelval.f ⊢ 𝐹 ∈ V
9 dicelval.s ⊢ 𝑆 ∈ V
10 1 2 3 4 5 6 7 dicval ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝐼 ‘ 𝑄 ) = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } )
11 10 eleq2d ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ⟨ 𝐹 , 𝑆 ⟩ ∈ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) )
12 eqeq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ↔ 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) )
13 12 anbi1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ) )
14 fveq1 ⊢ ( 𝑠 = 𝑆 → ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) )
15 14 eqeq2d ⊢ ( 𝑠 = 𝑆 → ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ↔ 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ) )
16 eleq1 ⊢ ( 𝑠 = 𝑆 → ( 𝑠 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸 ) )
17 15 16 anbi12d ⊢ ( 𝑠 = 𝑆 → ( ( 𝐹 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) )
18 8 9 13 17 opelopab ⊢ ( ⟨ 𝐹 , 𝑆 ⟩ ∈ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑠 ∈ 𝐸 ) } ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) )
19 11 18 bitrdi ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) )