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 ) ) ⟩ } ) )