| Step | Hyp | Ref
| Expression |
| 1 | | elex 3501 |
. 2
⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V) |
| 2 | | fveq2 6906 |
. . . . 5
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) |
| 3 | | erngset.h |
. . . . 5
⊢ 𝐻 = (LHyp‘𝐾) |
| 4 | 2, 3 | eqtr4di 2795 |
. . . 4
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) |
| 5 | | fveq2 6906 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (TEndo‘𝑘) = (TEndo‘𝐾)) |
| 6 | 5 | fveq1d 6908 |
. . . . . 6
⊢ (𝑘 = 𝐾 → ((TEndo‘𝑘)‘𝑤) = ((TEndo‘𝐾)‘𝑤)) |
| 7 | 6 | opeq2d 4880 |
. . . . 5
⊢ (𝑘 = 𝐾 → 〈(Base‘ndx),
((TEndo‘𝑘)‘𝑤)〉 = 〈(Base‘ndx),
((TEndo‘𝐾)‘𝑤)〉) |
| 8 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾)) |
| 9 | 8 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤)) |
| 10 | 9 | mpteq1d 5237 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) |
| 11 | 6, 6, 10 | mpoeq123dv 7508 |
. . . . . 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 4880 |
. . . . 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 → (𝑠 ∘ 𝑡) = (𝑠 ∘ 𝑡)) |
| 14 | 6, 6, 13 | mpoeq123dv 7508 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠 ∘ 𝑡))) |
| 15 | 14 | opeq2d 4880 |
. . . . 5
⊢ (𝑘 = 𝐾 → 〈(.r‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡))〉 = 〈(.r‘ndx),
(𝑠 ∈
((TEndo‘𝐾)‘𝑤), 𝑡 ∈ ((TEndo‘𝐾)‘𝑤) ↦ (𝑠 ∘ 𝑡))〉) |
| 16 | 7, 12, 15 | tpeq123d 4748 |
. . . 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) ↦ (𝑠 ∘ 𝑡))〉} = {〈(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) ↦ (𝑠 ∘ 𝑡))〉}) |
| 17 | 4, 16 | mpteq12dv 5233 |
. . 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) ↦ (𝑠 ∘ 𝑡))〉}) = (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) ↦ (𝑠 ∘ 𝑡))〉})) |
| 18 | | df-edring 40759 |
. . 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) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))〉, 〈(.r‘ndx),
(𝑠 ∈
((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡))〉})) |
| 19 | 17, 18, 3 | mptfvmpt 7248 |
. 2
⊢ (𝐾 ∈ V →
(EDRing‘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) ↦ (𝑠 ∘ 𝑡))〉})) |
| 20 | 1, 19 | syl 17 |
1
⊢ (𝐾 ∈ 𝑉 → (EDRing‘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) ↦ (𝑠 ∘ 𝑡))〉})) |