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Metamath Proof Explorer

Theorem dicfval

Description: The partial isomorphism C for a lattice K . (Contributed by NM, 15-Dec-2013)

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 )
Assertion dicfval ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 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 1 2 3 dicffval ⊢ ( 𝐾 ∈ 𝑉 → ( DIsoC ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) ) )
9 8 fveq1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) ) ‘ 𝑊 ) )
10 7 9 eqtrid ⊢ ( 𝐾 ∈ 𝑉 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) ) ‘ 𝑊 ) )
11 breq2 ⊢ ( 𝑤 = 𝑊 → ( 𝑟 ≤ 𝑤 ↔ 𝑟 ≤ 𝑊 ) )
12 11 notbid ⊢ ( 𝑤 = 𝑊 → ( ¬ 𝑟 ≤ 𝑤 ↔ ¬ 𝑟 ≤ 𝑊 ) )
13 12 rabbidv ⊢ ( 𝑤 = 𝑊 → { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } = { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } )
14 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) )
15 14 5 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) = 𝑇 )
16 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) = ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) )
17 16 4 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) = 𝑃 )
18 17 fveqeq2d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ↔ ( 𝑔 ‘ 𝑃 ) = 𝑞 ) )
19 15 18 riotaeqbidv ⊢ ( 𝑤 = 𝑊 → ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) = ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) )
20 19 fveq2d ⊢ ( 𝑤 = 𝑊 → ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) )
21 20 eqeq2d ⊢ ( 𝑤 = 𝑊 → ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ↔ 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ) )
22 fveq2 ⊢ ( 𝑤 = 𝑊 → ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) )
23 22 6 eqtr4di ⊢ ( 𝑤 = 𝑊 → ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) = 𝐸 )
24 23 eleq2d ⊢ ( 𝑤 = 𝑊 → ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↔ 𝑠 ∈ 𝐸 ) )
25 21 24 anbi12d ⊢ ( 𝑤 = 𝑊 → ( ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) ↔ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) ) )
26 25 opabbidv ⊢ ( 𝑤 = 𝑊 → { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } = { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) } )
27 13 26 mpteq12dv ⊢ ( 𝑤 = 𝑊 → ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) = ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) )
28 eqid ⊢ ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) ) = ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) )
29 2 fvexi ⊢ 𝐴 ∈ V
30 29 mptrabex ⊢ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) ∈ V
31 27 28 30 fvmpt ⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑤 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ( 𝑔 ‘ ( ( oc ‘ 𝐾 ) ‘ 𝑤 ) ) = 𝑞 ) ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ) } ) ) ‘ 𝑊 ) = ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) )
32 10 31 sylan9eq ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑞 ∈ { 𝑟 ∈ 𝐴 ∣ ¬ 𝑟 ≤ 𝑊 } ↦ { ⟨ 𝑓 , 𝑠 ⟩ ∣ ( 𝑓 = ( 𝑠 ‘ ( ℩ 𝑔 ∈ 𝑇 ( 𝑔 ‘ 𝑃 ) = 𝑞 ) ) ∧ 𝑠 ∈ 𝐸 ) } ) )