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Metamath Proof Explorer

Theorem dicopelval2

Description: Membership in value of the partial isomorphism C for a lattice K . (Contributed by NM, 20-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 )
dicval2.g ⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
dicelval2.f ⊢ 𝐹 ∈ V
dicelval2.s ⊢ 𝑆 ∈ V
Assertion dicopelval2 ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 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 dicval2.g ⊢ 𝐺 = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 )
9 dicelval2.f ⊢ 𝐹 ∈ V
10 dicelval2.s ⊢ 𝑆 ∈ V
11 1 2 3 4 5 6 7 9 10 dicopelval ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) ) )
12 8 fveq2i ⊢ ( 𝑆 ‘ 𝐺 ) = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) )
13 12 eqeq2i ⊢ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ↔ 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) )
14 13 anbi1i ⊢ ( ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ↔ ( 𝐹 = ( 𝑆 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑄 ) ) ∧ 𝑆 ∈ 𝐸 ) )
15 11 14 bitr4di ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ⟨ 𝐹 , 𝑆 ⟩ ∈ ( 𝐼 ‘ 𝑄 ) ↔ ( 𝐹 = ( 𝑆 ‘ 𝐺 ) ∧ 𝑆 ∈ 𝐸 ) ) )