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Theorem eengv 28730
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 6882 . . . . 5 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
21opeq2d 4873 . . . 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 7418 . . . . . . 7 ((𝑛 = 𝑁 ∧ (π‘₯ ∈ (π”Όβ€˜π‘›) ∧ 𝑦 ∈ (π”Όβ€˜π‘›))) β†’ (1...𝑛) = (1...𝑁))
65sumeq1d 15649 . . . . . 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 7476 . . . . 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 4873 . . . 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 4738 . . 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 3439 . . . . . 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 7476 . . . . 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 4873 . . . 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 4114 . . . . . 6 ((𝑛 = 𝑁 ∧ π‘₯ ∈ (π”Όβ€˜π‘›)) β†’ ((π”Όβ€˜π‘›) βˆ– {π‘₯}) = ((π”Όβ€˜π‘) βˆ– {π‘₯}))
151rabeqdv 3439 . . . . . . 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 7476 . . . . 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 4873 . . . 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 4738 . . 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 4157 . 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 28729 . 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 5423 . . 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 5423 . . 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 7727 . 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 6989 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 1083   = wceq 1533   ∈ wcel 2098  {crab 3424   βˆ– cdif 3938   βˆͺ cun 3939  {csn 4621  {cpr 4623  βŸ¨cop 4627   class class class wbr 5139  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  1c1 11108   βˆ’ cmin 11443  β„•cn 12211  2c2 12266  ...cfz 13485  β†‘cexp 14028  Ξ£csu 15634  ndxcnx 17131  Basecbs 17149  distcds 17211  Itvcitv 28177  LineGclng 28178  π”Όcee 28639   Btwn cbtwn 28640  EEGceeng 28728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-fun 6536  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seq 13968  df-sum 15635  df-eeng 28729
This theorem is referenced by:  eengstr  28731  eengbas  28732  ebtwntg  28733  ecgrtg  28734  elntg  28735
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