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Theorem erngset 40783
Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom 𝑊. (Contributed by NM, 5-Jun-2013.)
Hypotheses
Ref Expression
erngset.h 𝐻 = (LHyp‘𝐾)
erngset.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
erngset.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
erngset.d 𝐷 = ((EDRing‘𝐾)‘𝑊)
Assertion
Ref Expression
erngset ((𝐾𝑉𝑊𝐻) → 𝐷 = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩})
Distinct variable groups:   𝑓,𝑠,𝑡,𝐾   𝑓,𝑊,𝑠,𝑡
Allowed substitution hints:   𝐷(𝑡,𝑓,𝑠)   𝑇(𝑡,𝑓,𝑠)   𝐸(𝑡,𝑓,𝑠)   𝐻(𝑡,𝑓,𝑠)   𝑉(𝑡,𝑓,𝑠)

Proof of Theorem erngset
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 erngset.d . . 3 𝐷 = ((EDRing‘𝐾)‘𝑊)
2 erngset.h . . . . 5 𝐻 = (LHyp‘𝐾)
32erngfset 40782 . . . 4 (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))⟩}))
43fveq1d 6909 . . 3 (𝐾𝑉 → ((EDRing‘𝐾)‘𝑊) = ((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))⟩})‘𝑊))
51, 4eqtrid 2787 . 2 (𝐾𝑉𝐷 = ((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))⟩})‘𝑊))
6 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = ((TEndo‘𝐾)‘𝑊))
76opeq2d 4885 . . . . 5 (𝑤 = 𝑊 → ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩)
8 tpeq1 4747 . . . . . 6 (⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(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))⟩})
9 erngset.e . . . . . . . 8 𝐸 = ((TEndo‘𝐾)‘𝑊)
109opeq2i 4882 . . . . . . 7 ⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩
11 tpeq1 4747 . . . . . . 7 (⟨(Base‘ndx), 𝐸⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(Base‘ndx), 𝐸⟩, ⟨(+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))⟩})
1210, 11ax-mp 5 . . . . . 6 {⟨(Base‘ndx), 𝐸⟩, ⟨(+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))⟩}
138, 12eqtr4di 2793 . . . . 5 (⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑤)⟩ = ⟨(Base‘ndx), ((TEndo‘𝐾)‘𝑊)⟩ → {⟨(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), 𝐸⟩, ⟨(+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))⟩})
147, 13syl 17 . . . 4 (𝑤 = 𝑊 → {⟨(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), 𝐸⟩, ⟨(+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))⟩})
156, 9eqtr4di 2793 . . . . . . 7 (𝑤 = 𝑊 → ((TEndo‘𝐾)‘𝑤) = 𝐸)
16 fveq2 6907 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
17 erngset.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
1816, 17eqtr4di 2793 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
19 eqidd 2736 . . . . . . . 8 (𝑤 = 𝑊 → ((𝑠𝑓) ∘ (𝑡𝑓)) = ((𝑠𝑓) ∘ (𝑡𝑓)))
2018, 19mpteq12dv 5239 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))) = (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
2115, 15, 20mpoeq123dv 7508 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
2221opeq2d 4885 . . . . 5 (𝑤 = 𝑊 → ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩ = ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩)
2322tpeq2d 4751 . . . 4 (𝑤 = 𝑊 → {⟨(Base‘ndx), 𝐸⟩, ⟨(+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), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩})
24 eqidd 2736 . . . . . . 7 (𝑤 = 𝑊 → (𝑠𝑡) = (𝑠𝑡))
2515, 15, 24mpoeq123dv 7508 . . . . . 6 (𝑤 = 𝑊 → (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡)) = (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡)))
2625opeq2d 4885 . . . . 5 (𝑤 = 𝑊 → ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩ = ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩)
2726tpeq3d 4752 . . . 4 (𝑤 = 𝑊 → {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠𝑡))⟩} = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩})
2814, 23, 273eqtrd 2779 . . 3 (𝑤 = 𝑊 → {⟨(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), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩})
29 eqid 2735 . . 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))⟩}) = (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))⟩})
30 tpex 7765 . . 3 {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩} ∈ V
3128, 29, 30fvmpt 7016 . 2 (𝑊𝐻 → ((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))⟩})‘𝑊) = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩})
325, 31sylan9eq 2795 1 ((𝐾𝑉𝑊𝐻) → 𝐷 = {⟨(Base‘ndx), 𝐸⟩, ⟨(+g‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))⟩, ⟨(.r‘ndx), (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡))⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {ctp 4635  cop 4637  cmpt 5231  ccom 5693  cfv 6563  cmpo 7433  ndxcnx 17227  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  EDRingcedring 40736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-oprab 7435  df-mpo 7436  df-edring 40740
This theorem is referenced by:  erngbase  40784  erngfplus  40785  erngfmul  40788
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