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Theorem eengv 28789
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 6897 . . . . 5 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
21opeq2d 4881 . . . 4 (𝑛 = 𝑁 β†’ ⟨(Baseβ€˜ndx), (π”Όβ€˜π‘›)⟩ = ⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩)
31adantr 480 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘₯ ∈ (π”Όβ€˜π‘›)) β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
4 simpl 482 . . . . . . . 8 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ 𝑛 = 𝑁)
54oveq2d 7436 . . . . . . 7 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ (1...𝑛) = (1...𝑁))
65sumeq1d 15679 . . . . . 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 7494 . . . . 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 4881 . . . 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 4746 . . 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 480 . . . . . . 7 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
1110rabeqdv 3444 . . . . . 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 7494 . . . . 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 4881 . . . 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 4119 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘₯ ∈ (π”Όβ€˜π‘›)) β†’ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) = ((π”Όβ€˜π‘) βˆ– {π‘₯}))
151rabeqdv 3444 . . . . . . 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 480 . . . . . 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 7494 . . . . 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 4881 . . . 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 4746 . . 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 4163 . 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 28788 . 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 5434 . . 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 5434 . . 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 7748 . 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 7005 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 395   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099  {crab 3429   βˆ– cdif 3944   βˆͺ cun 3945  {csn 4629  {cpr 4631  βŸ¨cop 4635   class class class wbr 5148  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  1c1 11139   βˆ’ cmin 11474  β„•cn 12242  2c2 12297  ...cfz 13516  β†‘cexp 14058  Ξ£csu 15664  ndxcnx 17161  Basecbs 17179  distcds 17241  Itvcitv 28236  LineGclng 28237  π”Όcee 28698   Btwn cbtwn 28699  EEGceeng 28787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fun 6550  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-seq 13999  df-sum 15665  df-eeng 28788
This theorem is referenced by:  eengstr  28790  eengbas  28791  ebtwntg  28792  ecgrtg  28793  elntg  28794
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