This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem erngfset

Description: The division rings on trace-preserving endomorphisms for a lattice K . (Contributed by NM, 8-Jun-2013)

Ref Expression
Hypothesis erngset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion erngfset ( 𝐾 ∈ 𝑉 → ( EDRing ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) )

Proof

Step Hyp Ref Expression
1 erngset.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 elex ⊢ ( 𝐾 ∈ 𝑉 → 𝐾 ∈ V )
3 fveq2 ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = ( LHyp ‘ 𝐾 ) )
4 3 1 eqtr4di ⊢ ( 𝑘 = 𝐾 → ( LHyp ‘ 𝑘 ) = 𝐻 )
5 fveq2 ⊢ ( 𝑘 = 𝐾 → ( TEndo ‘ 𝑘 ) = ( TEndo ‘ 𝐾 ) )
6 5 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) )
7 6 opeq2d ⊢ ( 𝑘 = 𝐾 → ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ⟩ = ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ )
8 fveq2 ⊢ ( 𝑘 = 𝐾 → ( LTrn ‘ 𝑘 ) = ( LTrn ‘ 𝐾 ) )
9 8 fveq1d ⊢ ( 𝑘 = 𝐾 → ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) = ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) )
10 9 mpteq1d ⊢ ( 𝑘 = 𝐾 → ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) = ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) )
11 6 6 10 mpoeq123dv ⊢ ( 𝑘 = 𝐾 → ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) )
12 11 opeq2d ⊢ ( 𝑘 = 𝐾 → ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ = ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ )
13 eqidd ⊢ ( 𝑘 = 𝐾 → ( 𝑠 ∘ 𝑡 ) = ( 𝑠 ∘ 𝑡 ) )
14 6 6 13 mpoeq123dv ⊢ ( 𝑘 = 𝐾 → ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) = ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) )
15 14 opeq2d ⊢ ( 𝑘 = 𝐾 → ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ = ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ )
16 7 12 15 tpeq123d ⊢ ( 𝑘 = 𝐾 → { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } = { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } )
17 4 16 mpteq12dv ⊢ ( 𝑘 = 𝐾 → ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) )
18 df-edring ⊢ EDRing = ( 𝑘 ∈ V ↦ ( 𝑤 ∈ ( LHyp ‘ 𝑘 ) ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝑘 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) )
19 17 18 1 mptfvmpt ⊢ ( 𝐾 ∈ V → ( EDRing ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) )
20 2 19 syl ⊢ ( 𝐾 ∈ 𝑉 → ( EDRing ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { ⟨ ( Base ‘ ndx ) , ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ⟩ , ⟨ ( +g ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑓 ∈ ( ( LTrn ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( ( 𝑠 ‘ 𝑓 ) ∘ ( 𝑡 ‘ 𝑓 ) ) ) ) ⟩ , ⟨ ( .r ‘ ndx ) , ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) , 𝑡 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑤 ) ↦ ( 𝑠 ∘ 𝑡 ) ) ⟩ } ) )