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Theorem eengv 27970
Description: The value of the Euclidean geometry for dimension 𝑁. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Assertion
Ref Expression
eengv (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) = ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
Distinct variable group:   π‘₯,𝑖,𝑦,𝑧,𝑁

Proof of Theorem eengv
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6843 . . . . 5 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
21opeq2d 4838 . . . 4 (𝑛 = 𝑁 β†’ ⟨(Baseβ€˜ndx), (π”Όβ€˜π‘›)⟩ = ⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩)
31adantr 482 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘₯ ∈ (π”Όβ€˜π‘›)) β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
4 simpl 484 . . . . . . . 8 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ 𝑛 = 𝑁)
54oveq2d 7374 . . . . . . 7 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ (1...𝑛) = (1...𝑁))
65sumeq1d 15591 . . . . . 6 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2) = Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))
71, 3, 6mpoeq123dva 7432 . . . . 5 (𝑛 = 𝑁 β†’ (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2)) = (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2)))
87opeq2d 4838 . . . 4 (𝑛 = 𝑁 β†’ ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩ = ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩)
92, 8preq12d 4703 . . 3 (𝑛 = 𝑁 β†’ {⟨(Baseβ€˜ndx), (π”Όβ€˜π‘›)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} = {⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩})
101adantr 482 . . . . . . 7 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
1110rabeqdv 3421 . . . . . 6 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©} = {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})
121, 3, 11mpoeq123dva 7432 . . . . 5 (𝑛 = 𝑁 β†’ (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}) = (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}))
1312opeq2d 4838 . . . 4 (𝑛 = 𝑁 β†’ ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩ = ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩)
143difeq1d 4082 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘₯ ∈ (π”Όβ€˜π‘›)) β†’ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) = ((π”Όβ€˜π‘) βˆ– {π‘₯}))
151rabeqdv 3421 . . . . . . 7 (𝑛 = 𝑁 β†’ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)} = {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})
1615adantr 482 . . . . . 6 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}))) β†’ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)} = {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})
171, 14, 16mpoeq123dva 7432 . . . . 5 (𝑛 = 𝑁 β†’ (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) = (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}))
1817opeq2d 4838 . . . 4 (𝑛 = 𝑁 β†’ ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ = ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩)
1913, 18preq12d 4703 . . 3 (𝑛 = 𝑁 β†’ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩} = {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩})
209, 19uneq12d 4125 . 2 (𝑛 = 𝑁 β†’ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘›)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}) = ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
21 df-eeng 27969 . 2 EEG = (𝑛 ∈ β„• ↦ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘›)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ Σ𝑖 ∈ (1...𝑛)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ (π”Όβ€˜π‘›) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘›), 𝑦 ∈ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘›) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
22 prex 5390 . . 3 {⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} ∈ V
23 prex 5390 . . 3 {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩} ∈ V
2422, 23unex 7681 . 2 ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}) ∈ V
2520, 21, 24fvmpt 6949 1 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) = ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  {crab 3406   βˆ– cdif 3908   βˆͺ cun 3909  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1c1 11057   βˆ’ cmin 11390  β„•cn 12158  2c2 12213  ...cfz 13430  β†‘cexp 13973  Ξ£csu 15576  ndxcnx 17070  Basecbs 17088  distcds 17147  Itvcitv 27417  LineGclng 27418  π”Όcee 27879   Btwn cbtwn 27880  EEGceeng 27968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seq 13913  df-sum 15577  df-eeng 27969
This theorem is referenced by:  eengstr  27971  eengbas  27972  ebtwntg  27973  ecgrtg  27974  elntg  27975
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