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Theorem erngfset 40328
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 3482 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6892 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 erngset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2783 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6892 . . . . . . 7 (π‘˜ = 𝐾 β†’ (TEndoβ€˜π‘˜) = (TEndoβ€˜πΎ))
65fveq1d 6894 . . . . . 6 (π‘˜ = 𝐾 β†’ ((TEndoβ€˜π‘˜)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘€))
76opeq2d 4876 . . . . 5 (π‘˜ = 𝐾 β†’ ⟨(Baseβ€˜ndx), ((TEndoβ€˜π‘˜)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩)
8 fveq2 6892 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
98fveq1d 6894 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
109mpteq1d 5238 . . . . . . 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 7492 . . . . . 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 4876 . . . . 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 2726 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑠 ∘ 𝑑) = (𝑠 ∘ 𝑑))
146, 6, 13mpoeq123dv 7492 . . . . . 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 4876 . . . . 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 4748 . . . 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 5234 . . 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 40286 . . 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 7236 . 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 1533   ∈ wcel 2098  Vcvv 3463  {ctp 4628  βŸ¨cop 4630   ↦ cmpt 5226   ∘ ccom 5676  β€˜cfv 6543   ∈ cmpo 7418  ndxcnx 17161  Basecbs 17179  +gcplusg 17232  .rcmulr 17233  LHypclh 39513  LTrncltrn 39630  TEndoctendo 40281  EDRingcedring 40282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-oprab 7420  df-mpo 7421  df-edring 40286
This theorem is referenced by:  erngset  40329
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