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Theorem istrkgb 28477
Description: Property of being a Tarski geometry - betweenness part. (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
istrkgb (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
Distinct variable groups:   𝑎,𝑏,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧,𝐼   𝑃,𝑎,𝑏,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   ,𝑎,𝑏,𝑢,𝑣,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑣,𝑢,𝑡,𝑠,𝑎,𝑏)   (𝑡,𝑠)

Proof of Theorem istrkgb
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 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
4 simpr 484 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
54oveqd 7447 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑥) = (𝑥𝐼𝑥))
65eleq2d 2824 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝑖𝑥) ↔ 𝑦 ∈ (𝑥𝐼𝑥)))
76imbi1d 341 . . . . . 6 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ↔ (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦)))
83, 7raleqbidv 3343 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ↔ ∀𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦)))
93, 8raleqbidv 3343 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ↔ ∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦)))
104oveqd 7447 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑧) = (𝑥𝐼𝑧))
1110eleq2d 2824 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑢 ∈ (𝑥𝑖𝑧) ↔ 𝑢 ∈ (𝑥𝐼𝑧)))
124oveqd 7447 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑦𝑖𝑧) = (𝑦𝐼𝑧))
1312eleq2d 2824 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑣 ∈ (𝑦𝑖𝑧) ↔ 𝑣 ∈ (𝑦𝐼𝑧)))
1411, 13anbi12d 632 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) ↔ (𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧))))
154oveqd 7447 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑢𝑖𝑦) = (𝑢𝐼𝑦))
1615eleq2d 2824 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎 ∈ (𝑢𝑖𝑦) ↔ 𝑎 ∈ (𝑢𝐼𝑦)))
174oveqd 7447 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑣𝑖𝑥) = (𝑣𝐼𝑥))
1817eleq2d 2824 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎 ∈ (𝑣𝑖𝑥) ↔ 𝑎 ∈ (𝑣𝐼𝑥)))
1916, 18anbi12d 632 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)) ↔ (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
203, 19rexeqbidv 3344 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥)) ↔ ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))))
2114, 20imbi12d 344 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
223, 21raleqbidv 3343 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ∀𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
233, 22raleqbidv 3343 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ∀𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
243, 23raleqbidv 3343 . . . . . 6 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ∀𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
253, 24raleqbidv 3343 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ∀𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
263, 25raleqbidv 3343 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ↔ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥)))))
273pweqd 4621 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝒫 𝑝 = 𝒫 𝑃)
284oveqd 7447 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎𝑖𝑦) = (𝑎𝐼𝑦))
2928eleq2d 2824 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥 ∈ (𝑎𝑖𝑦) ↔ 𝑥 ∈ (𝑎𝐼𝑦)))
30292ralbidv 3218 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) ↔ ∀𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
313, 30rexeqbidv 3344 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) ↔ ∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦)))
324oveqd 7447 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑥𝑖𝑦) = (𝑥𝐼𝑦))
3332eleq2d 2824 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏 ∈ (𝑥𝑖𝑦) ↔ 𝑏 ∈ (𝑥𝐼𝑦)))
34332ralbidv 3218 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦) ↔ ∀𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
353, 34rexeqbidv 3344 . . . . . . 7 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦) ↔ ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))
3631, 35imbi12d 344 . . . . . 6 ((𝑝 = 𝑃𝑖 = 𝐼) → ((∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)) ↔ (∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
3727, 36raleqbidv 3343 . . . . 5 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)) ↔ ∀𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
3827, 37raleqbidv 3343 . . . 4 ((𝑝 = 𝑃𝑖 = 𝐼) → (∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)) ↔ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦))))
399, 26, 383anbi123d 1435 . . 3 ((𝑝 = 𝑃𝑖 = 𝐼) → ((∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦))) ↔ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
401, 2, 39sbcie2s 17194 . 2 (𝑓 = 𝐺 → ([(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦))) ↔ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
41 df-trkgb 28471 . 2 TarskiGB = {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](∀𝑥𝑝𝑦𝑝 (𝑦 ∈ (𝑥𝑖𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑝𝑦𝑝𝑧𝑝𝑢𝑝𝑣𝑝 ((𝑢 ∈ (𝑥𝑖𝑧) ∧ 𝑣 ∈ (𝑦𝑖𝑧)) → ∃𝑎𝑝 (𝑎 ∈ (𝑢𝑖𝑦) ∧ 𝑎 ∈ (𝑣𝑖𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑝𝑡 ∈ 𝒫 𝑝(∃𝑎𝑝𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝑖𝑦) → ∃𝑏𝑝𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝑖𝑦)))}
4240, 41elab4g 3685 1 (𝐺 ∈ TarskiGB ↔ (𝐺 ∈ V ∧ (∀𝑥𝑃𝑦𝑃 (𝑦 ∈ (𝑥𝐼𝑥) → 𝑥 = 𝑦) ∧ ∀𝑥𝑃𝑦𝑃𝑧𝑃𝑢𝑃𝑣𝑃 ((𝑢 ∈ (𝑥𝐼𝑧) ∧ 𝑣 ∈ (𝑦𝐼𝑧)) → ∃𝑎𝑃 (𝑎 ∈ (𝑢𝐼𝑦) ∧ 𝑎 ∈ (𝑣𝐼𝑥))) ∧ ∀𝑠 ∈ 𝒫 𝑃𝑡 ∈ 𝒫 𝑃(∃𝑎𝑃𝑥𝑠𝑦𝑡 𝑥 ∈ (𝑎𝐼𝑦) → ∃𝑏𝑃𝑥𝑠𝑦𝑡 𝑏 ∈ (𝑥𝐼𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477  [wsbc 3790  𝒫 cpw 4604  cfv 6562  (class class class)co 7430  Basecbs 17244  distcds 17306  TarskiGBcstrkgb 28451  Itvcitv 28455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-trkgb 28471
This theorem is referenced by:  axtgbtwnid  28488  axtgpasch  28489  axtgcont1  28490  f1otrg  28893  eengtrkg  29015
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