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Theorem erngfset 36607
Description: The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
Hypothesis
Ref Expression
erngset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
erngfset (𝐾𝑉 → (EDRing‘𝐾) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩}))
Distinct variable groups:   𝑤,𝐻   𝑓,𝑠,𝑡,𝑤,𝐾
Allowed substitution hints:   𝐻(𝑡,𝑓,𝑠)   𝑉(𝑤,𝑡,𝑓,𝑠)

Proof of Theorem erngfset
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6353 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 erngset.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2812 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6353 . . . . . . 7 (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
65fveq1d 6355 . . . . . 6 (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
76opeq2d 4560 . . . . 5 (𝑘 = 𝐾 → ⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩)
8 fveq2 6353 . . . . . . . . 9 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
98fveq1d 6355 . . . . . . . 8 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
109mpteq1d 4890 . . . . . . 7 (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
116, 6, 10mpt2eq123dv 6883 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
1211opeq2d 4560 . . . . 5 (𝑘 = 𝐾 → ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩ = ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩)
13 eqidd 2761 . . . . . . 7 (𝑘 = 𝐾 → (𝑠𝑡) = (𝑠𝑡))
146, 6, 13mpt2eq123dv 6883 . . . . . 6 (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠𝑡)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡)))
1514opeq2d 4560 . . . . 5 (𝑘 = 𝐾 → ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠𝑡))⟩ = ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩)
167, 12, 15tpeq123d 4427 . . . 4 (𝑘 = 𝐾 → {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠𝑡))⟩} = {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩})
174, 16mpteq12dv 4885 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠𝑡))⟩}) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩}))
18 df-edring 36565 . . 3 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) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠𝑡))⟩}))
19 fvex 6363 . . . . 5 (LHyp‘𝐾) ∈ V
203, 19eqeltri 2835 . . . 4 𝐻 ∈ V
2120mptex 6651 . . 3 (𝑤𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩}) ∈ V
2217, 18, 21fvmpt 6445 . 2 (𝐾 ∈ V → (EDRing‘𝐾) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩}))
231, 22syl 17 1 (𝐾𝑉 → (EDRing‘𝐾) = (𝑤𝐻 ↦ {⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  {ctp 4325  cop 4327  cmpt 4881  ccom 5270  cfv 6049  cmpt2 6816  ndxcnx 16076  Basecbs 16079  +gcplusg 16163  .rcmulr 16164  LHypclh 35791  LTrncltrn 35908  TEndoctendo 36560  EDRingcedring 36561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-oprab 6818  df-mpt2 6819  df-edring 36565
This theorem is referenced by:  erngset  36608
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