MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  istrkge Structured version   Visualization version   GIF version

Theorem istrkge 28480
Description: Property of fulfilling Euclid's axiom. (Contributed by Thierry Arnoux, 14-Mar-2019.)
Hypotheses
Ref Expression
istrkg.p 𝑃 = (Base‘𝐺)
istrkg.d = (dist‘𝐺)
istrkg.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
istrkge (𝐺 ∈ TarskiGE ↔ (𝐺 ∈ V ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
Distinct variable groups:   𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧,𝐼   𝑃,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧   ,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑣,𝑢,𝑎,𝑏)

Proof of Theorem istrkge
Dummy variables 𝑓 𝑖 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg.p . . 3 𝑃 = (Base‘𝐺)
2 istrkg.i . . 3 𝐼 = (Itv‘𝐺)
3 simpl 482 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
4 simpr 484 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
54oveqd 7448 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑣) = (𝑥𝐼𝑣))
65eleq2d 2825 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑢 ∈ (𝑥𝑖𝑣) ↔ 𝑢 ∈ (𝑥𝐼𝑣)))
74oveqd 7448 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑦𝑖𝑧) = (𝑦𝐼𝑧))
87eleq2d 2825 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑢 ∈ (𝑦𝑖𝑧) ↔ 𝑢 ∈ (𝑦𝐼𝑧)))
96, 83anbi12d 1436 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) ↔ (𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢)))
104oveqd 7448 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑎) = (𝑥𝐼𝑎))
1110eleq2d 2825 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝑖𝑎) ↔ 𝑦 ∈ (𝑥𝐼𝑎)))
124oveqd 7448 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑏) = (𝑥𝐼𝑏))
1312eleq2d 2825 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝑖𝑏) ↔ 𝑧 ∈ (𝑥𝐼𝑏)))
144oveqd 7448 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎𝑖𝑏) = (𝑎𝐼𝑏))
1514eleq2d 2825 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑣 ∈ (𝑎𝑖𝑏) ↔ 𝑣 ∈ (𝑎𝐼𝑏)))
1611, 13, 153anbi123d 1435 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)) ↔ (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏))))
173, 16rexeqbidv 3345 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)) ↔ ∃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏))))
183, 17rexeqbidv 3345 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)) ↔ ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏))))
199, 18imbi12d 344 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
203, 19raleqbidv 3344 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
213, 20raleqbidv 3344 . . . . . 6 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
223, 21raleqbidv 3344 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
233, 22raleqbidv 3344 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
243, 23raleqbidv 3344 . . 3 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
251, 2, 24sbcie2s 17195 . 2 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏))) ↔ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
26 df-trkge 28474 . 2 TarskiGE = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖]𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑣) ∧ 𝑢 ∈ (𝑦𝑖𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑝𝑏𝑝 (𝑦 ∈ (𝑥𝑖𝑎) ∧ 𝑧 ∈ (𝑥𝑖𝑏) ∧ 𝑣 ∈ (𝑎𝑖𝑏)))}
2725, 26elab4g 3686 1 (𝐺 ∈ TarskiGE ↔ (𝐺 ∈ V ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑣) ∧ 𝑢 ∈ (𝑦𝐼𝑧) ∧ 𝑥𝑢) → ∃𝑎𝑃𝑏𝑃 (𝑦 ∈ (𝑥𝐼𝑎) ∧ 𝑧 ∈ (𝑥𝐼𝑏) ∧ 𝑣 ∈ (𝑎𝐼𝑏)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  [wsbc 3791  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGEcstrkge 28455  Itvcitv 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkge 28474
This theorem is referenced by:  axtgeucl  28495  f1otrge  28895  eengtrkge  29017
  Copyright terms: Public domain W3C validator