Theorem erngfset-rN 41264

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

Hypothesis

Ref	Expression
erngset.h-r	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
erngfset-rN	⊢ (𝐾 ∈ 𝑉 → (EDRingR‘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) ↦ (𝑡 ∘ 𝑠))⟩}))

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-rN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3451	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
2		fveq2 6832	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		erngset.h-r	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6832	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾))
6	5	fveq1d 6834	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤))
7	6	opeq2d 4824	. . . . 5 ⊢ (𝑘 = 𝐾 → ⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩)
8		fveq2 6832	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
9	8	fveq1d 6834	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
10	9	mpteq1d 5176	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
11	6, 6, 10	mpoeq123dv 7433	. . . . . 6 ⊢ (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 4824	. . . . 5 ⊢ (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 2738	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑡 ∘ 𝑠) = (𝑡 ∘ 𝑠))
14	6, 6, 13	mpoeq123dv 7433	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠)))
15	14	opeq2d 4824	. . . . 5 ⊢ (𝑘 = 𝐾 → ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩ = ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩)
16	7, 12, 15	tpeq123d 4693	. . . 4 ⊢ (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) ↦ (𝑡 ∘ 𝑠))⟩} = {⟨(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) ↦ (𝑡 ∘ 𝑠))⟩})
17	4, 16	mpteq12dv 5173	. . 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) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}) = (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) ↦ (𝑡 ∘ 𝑠))⟩}))
18		df-edring-rN 41213	. . 3 ⊢ EDRingR = (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) ↦ (𝑡 ∘ 𝑠))⟩}))
19	17, 18, 3	mptfvmpt 7174	. 2 ⊢ (𝐾 ∈ V → (EDRingR‘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) ↦ (𝑡 ∘ 𝑠))⟩}))
20	1, 19	syl 17	1 ⊢ (𝐾 ∈ 𝑉 → (EDRingR‘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) ↦ (𝑡 ∘ 𝑠))⟩}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {ctp 4572  ⟨cop 4574   ↦ cmpt 5167   ∘ ccom 5626  ‘cfv 6490   ∈ cmpo 7360  ndxcnx 17152  Basecbs 17168  +_gcplusg 17209  ._rcmulr 17210  LHypclh 40441  LTrncltrn 40558  TEndoctendo 41209  EDRing_Rcedring-rN 41211

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-oprab 7362  df-mpo 7363  df-edring-rN 41213

This theorem is referenced by:  erngset-rN  41265