Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  erngset-rN Structured version   Visualization version   GIF version

Theorem erngset-rN 39983
Description: The division ring on trace-preserving endomorphisms for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
erngset.h-r 𝐻 = (LHypβ€˜πΎ)
erngset.t-r 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
erngset.e-r 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
erngset.d-r 𝐷 = ((EDRingRβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
erngset-rN ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
Distinct variable groups:   𝑓,𝑠,𝑑,𝐾   𝑓,π‘Š,𝑠,𝑑
Allowed substitution hints:   𝐷(𝑑,𝑓,𝑠)   𝑇(𝑑,𝑓,𝑠)   𝐸(𝑑,𝑓,𝑠)   𝐻(𝑑,𝑓,𝑠)   𝑉(𝑑,𝑓,𝑠)

Proof of Theorem erngset-rN
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 erngset.d-r . . 3 𝐷 = ((EDRingRβ€˜πΎ)β€˜π‘Š)
2 erngset.h-r . . . . 5 𝐻 = (LHypβ€˜πΎ)
32erngfset-rN 39982 . . . 4 (𝐾 ∈ 𝑉 β†’ (EDRingRβ€˜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‘€) ↦ (𝑑 ∘ 𝑠))⟩}))
43fveq1d 6893 . . 3 (𝐾 ∈ 𝑉 β†’ ((EDRingRβ€˜πΎ)β€˜π‘Š) = ((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‘€) ↦ (𝑑 ∘ 𝑠))⟩})β€˜π‘Š))
51, 4eqtrid 2783 . 2 (𝐾 ∈ 𝑉 β†’ 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‘€) ↦ (𝑑 ∘ 𝑠))⟩})β€˜π‘Š))
6 fveq2 6891 . . . . . 6 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = ((TEndoβ€˜πΎ)β€˜π‘Š))
76opeq2d 4880 . . . . 5 (𝑀 = π‘Š β†’ ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩)
8 tpeq1 4746 . . . . . 6 (⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩ β†’ {⟨(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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩})
9 erngset.e-r . . . . . . . 8 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
109opeq2i 4877 . . . . . . 7 ⟨(Baseβ€˜ndx), 𝐸⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩
11 tpeq1 4746 . . . . . . 7 (⟨(Baseβ€˜ndx), 𝐸⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩ β†’ {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩})
1210, 11ax-mp 5 . . . . . 6 {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩}
138, 12eqtr4di 2789 . . . . 5 (⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩ = ⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘Š)⟩ β†’ {⟨(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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩})
147, 13syl 17 . . . 4 (𝑀 = π‘Š β†’ {⟨(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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩})
156, 9eqtr4di 2789 . . . . . . 7 (𝑀 = π‘Š β†’ ((TEndoβ€˜πΎ)β€˜π‘€) = 𝐸)
16 fveq2 6891 . . . . . . . . 9 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
17 erngset.t-r . . . . . . . . 9 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
1816, 17eqtr4di 2789 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = 𝑇)
19 eqidd 2732 . . . . . . . 8 (𝑀 = π‘Š β†’ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)) = ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))
2018, 19mpteq12dv 5239 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))) = (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))
2115, 15, 20mpoeq123dv 7487 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))) = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
2221opeq2d 4880 . . . . 5 (𝑀 = π‘Š β†’ ⟨(+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‘“))))⟩ = ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩)
2322tpeq2d 4750 . . . 4 (𝑀 = π‘Š β†’ {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩})
24 eqidd 2732 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑑 ∘ 𝑠) = (𝑑 ∘ 𝑠))
2515, 15, 24mpoeq123dv 7487 . . . . . 6 (𝑀 = π‘Š β†’ (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠)) = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠)))
2625opeq2d 4880 . . . . 5 (𝑀 = π‘Š β†’ ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩ = ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩)
2726tpeq3d 4751 . . . 4 (𝑀 = π‘Š β†’ {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
2814, 23, 273eqtrd 2775 . . 3 (𝑀 = π‘Š β†’ {⟨(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‘€) ↦ (𝑑 ∘ 𝑠))⟩} = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
29 eqid 2731 . . 3 (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‘€) ↦ (𝑑 ∘ 𝑠))⟩}) = (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‘€) ↦ (𝑑 ∘ 𝑠))⟩})
30 tpex 7738 . . 3 {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩} ∈ V
3128, 29, 30fvmpt 6998 . 2 (π‘Š ∈ 𝐻 β†’ ((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‘€) ↦ (𝑑 ∘ 𝑠))⟩})β€˜π‘Š) = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
325, 31sylan9eq 2791 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {ctp 4632  βŸ¨cop 4634   ↦ cmpt 5231   ∘ ccom 5680  β€˜cfv 6543   ∈ cmpo 7414  ndxcnx 17131  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  LHypclh 39159  LTrncltrn 39276  TEndoctendo 39927  EDRingRcedring-rN 39929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  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 7416  df-mpo 7417  df-edring-rN 39931
This theorem is referenced by:  erngbase-rN  39984  erngfplus-rN  39985  erngfmul-rN  39988
  Copyright terms: Public domain W3C validator