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Theorem ebtwntg 28813
Description: The betweenness relation used in the Tarski structure for the Euclidean geometry is the same as Btwn. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
ebtwntg.1 (πœ‘ β†’ 𝑁 ∈ β„•)
ebtwntg.2 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
ebtwntg.3 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
ebtwntg.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
ebtwntg.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
ebtwntg.z (πœ‘ β†’ 𝑍 ∈ 𝑃)
Assertion
Ref Expression
ebtwntg (πœ‘ β†’ (𝑍 Btwn βŸ¨π‘‹, π‘ŒβŸ© ↔ 𝑍 ∈ (π‘‹πΌπ‘Œ)))

Proof of Theorem ebtwntg
Dummy variables π‘₯ 𝑖 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ebtwntg.3 . . . . 5 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
2 itvid 28263 . . . . . 6 Itv = Slot (Itvβ€˜ndx)
3 fvexd 6917 . . . . . 6 (πœ‘ β†’ (EEGβ€˜π‘) ∈ V)
4 ebtwntg.1 . . . . . . . . 9 (πœ‘ β†’ 𝑁 ∈ β„•)
5 eengstr 28811 . . . . . . . . 9 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
64, 5syl 17 . . . . . . . 8 (πœ‘ β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
7 structn0fun 17127 . . . . . . . 8 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
86, 7syl 17 . . . . . . 7 (πœ‘ β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
9 structcnvcnv 17129 . . . . . . . . 9 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
106, 9syl 17 . . . . . . . 8 (πœ‘ β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
1110funeqd 6580 . . . . . . 7 (πœ‘ β†’ (Fun β—‘β—‘(EEGβ€˜π‘) ↔ Fun ((EEGβ€˜π‘) βˆ– {βˆ…})))
128, 11mpbird 256 . . . . . 6 (πœ‘ β†’ Fun β—‘β—‘(EEGβ€˜π‘))
13 opex 5470 . . . . . . . . 9 ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩ ∈ V
1413prid1 4771 . . . . . . . 8 ⟨(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©})⟩, ⟨(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©)})⟩}
15 elun2 4179 . . . . . . . 8 (⟨(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©})⟩, ⟨(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©)})⟩} β†’ ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 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©)})⟩}))
1614, 15ax-mp 5 . . . . . . 7 ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 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©)})⟩})
17 eengv 28810 . . . . . . . 8 (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©)})⟩}))
184, 17syl 17 . . . . . . 7 (πœ‘ 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©)})⟩}))
1916, 18eleqtrrid 2836 . . . . . 6 (πœ‘ β†’ ⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩ ∈ (EEGβ€˜π‘))
20 fvex 6915 . . . . . . . 8 (π”Όβ€˜π‘) ∈ V
2120, 20mpoex 8090 . . . . . . 7 (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}) ∈ V
2221a1i 11 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}) ∈ V)
232, 3, 12, 19, 22strfv2d 17178 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}) = (Itvβ€˜(EEGβ€˜π‘)))
241, 23eqtr4id 2787 . . . 4 (πœ‘ β†’ 𝐼 = (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©}))
25 simprl 769 . . . . . . 7 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ π‘₯ = 𝑋)
26 simprr 771 . . . . . . 7 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ 𝑦 = π‘Œ)
2725, 26opeq12d 4886 . . . . . 6 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ ⟨π‘₯, π‘¦βŸ© = βŸ¨π‘‹, π‘ŒβŸ©)
2827breq2d 5164 . . . . 5 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ↔ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©))
2928rabbidv 3438 . . . 4 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©} = {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©})
30 ebtwntg.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑃)
31 ebtwntg.2 . . . . . 6 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
3230, 31eleqtrdi 2839 . . . . 5 (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜(EEGβ€˜π‘)))
33 eengbas 28812 . . . . . 6 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
344, 33syl 17 . . . . 5 (πœ‘ β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
3532, 34eleqtrrd 2832 . . . 4 (πœ‘ β†’ 𝑋 ∈ (π”Όβ€˜π‘))
36 ebtwntg.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑃)
3736, 31eleqtrdi 2839 . . . . 5 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜(EEGβ€˜π‘)))
3837, 34eleqtrrd 2832 . . . 4 (πœ‘ β†’ π‘Œ ∈ (π”Όβ€˜π‘))
3920rabex 5338 . . . . 5 {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©} ∈ V
4039a1i 11 . . . 4 (πœ‘ β†’ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©} ∈ V)
4124, 29, 35, 38, 40ovmpod 7579 . . 3 (πœ‘ β†’ (π‘‹πΌπ‘Œ) = {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©})
4241eleq2d 2815 . 2 (πœ‘ β†’ (𝑍 ∈ (π‘‹πΌπ‘Œ) ↔ 𝑍 ∈ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©}))
43 ebtwntg.z . . . . 5 (πœ‘ β†’ 𝑍 ∈ 𝑃)
4443, 31eleqtrdi 2839 . . . 4 (πœ‘ β†’ 𝑍 ∈ (Baseβ€˜(EEGβ€˜π‘)))
4544, 34eleqtrrd 2832 . . 3 (πœ‘ β†’ 𝑍 ∈ (π”Όβ€˜π‘))
46 breq1 5155 . . . 4 (𝑧 = 𝑍 β†’ (𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ© ↔ 𝑍 Btwn βŸ¨π‘‹, π‘ŒβŸ©))
4746elrab3 3685 . . 3 (𝑍 ∈ (π”Όβ€˜π‘) β†’ (𝑍 ∈ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©} ↔ 𝑍 Btwn βŸ¨π‘‹, π‘ŒβŸ©))
4845, 47syl 17 . 2 (πœ‘ β†’ (𝑍 ∈ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn βŸ¨π‘‹, π‘ŒβŸ©} ↔ 𝑍 Btwn βŸ¨π‘‹, π‘ŒβŸ©))
4942, 48bitr2d 279 1 (πœ‘ β†’ (𝑍 Btwn βŸ¨π‘‹, π‘ŒβŸ© ↔ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  {crab 3430  Vcvv 3473   βˆ– cdif 3946   βˆͺ cun 3947  βˆ…c0 4326  {csn 4632  {cpr 4634  βŸ¨cop 4638   class class class wbr 5152  β—‘ccnv 5681  Fun wfun 6547  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1c1 11147   βˆ’ cmin 11482  β„•cn 12250  2c2 12305  7c7 12310  cdc 12715  ...cfz 13524  β†‘cexp 14066  Ξ£csu 15672   Struct cstr 17122  ndxcnx 17169  Basecbs 17187  distcds 17249  Itvcitv 28257  LineGclng 28258  π”Όcee 28719   Btwn cbtwn 28720  EEGceeng 28808
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-seq 14007  df-sum 15673  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-ds 17262  df-itv 28259  df-lng 28260  df-eeng 28809
This theorem is referenced by:  elntg  28815  elntg2  28816  eengtrkg  28817  eengtrkge  28818
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