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Theorem erngfset 39229
Description: The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
Hypothesis
Ref Expression
erngset.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
erngfset (𝐾 ∈ 𝑉 β†’ (EDRingβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
Distinct variable groups:   𝑀,𝐻   𝑓,𝑠,𝑑,𝑀,𝐾
Allowed substitution hints:   𝐻(𝑑,𝑓,𝑠)   𝑉(𝑀,𝑑,𝑓,𝑠)

Proof of Theorem erngfset
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3461 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6839 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 erngset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2794 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6839 . . . . . . 7 (π‘˜ = 𝐾 β†’ (TEndoβ€˜π‘˜) = (TEndoβ€˜πΎ))
65fveq1d 6841 . . . . . 6 (π‘˜ = 𝐾 β†’ ((TEndoβ€˜π‘˜)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘€))
76opeq2d 4835 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩)
8 fveq2 6839 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
98fveq1d 6841 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
109mpteq1d 5198 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))
116, 6, 10mpoeq123dv 7428 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
1211opeq2d 4835 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩ = ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩)
13 eqidd 2737 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑠 ∘ 𝑑) = (𝑠 ∘ 𝑑))
146, 6, 13mpoeq123dv 7428 . . . . . 6 (π‘˜ = 𝐾 β†’ (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑)) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑)))
1514opeq2d 4835 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩ = ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩)
167, 12, 15tpeq123d 4707 . . . 4 (π‘˜ = 𝐾 β†’ {⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩} = {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩})
174, 16mpteq12dv 5194 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
18 df-edring 39187 . . 3 EDRing = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜π‘˜)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
1917, 18, 3mptfvmpt 7174 . 2 (𝐾 ∈ V β†’ (EDRingβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
201, 19syl 17 1 (𝐾 ∈ 𝑉 β†’ (EDRingβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443  {ctp 4588  βŸ¨cop 4590   ↦ cmpt 5186   ∘ ccom 5635  β€˜cfv 6493   ∈ cmpo 7355  ndxcnx 17057  Basecbs 17075  +gcplusg 17125  .rcmulr 17126  LHypclh 38414  LTrncltrn 38531  TEndoctendo 39182  EDRingcedring 39183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-oprab 7357  df-mpo 7358  df-edring 39187
This theorem is referenced by:  erngset  39230
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