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Theorem istrkgld 25158
 Description: Property of fulfilling the lower dimension 𝑁 axiom. (Contributed by Thierry Arnoux, 20-Nov-2019.)
Hypotheses
Ref Expression
istrkg.p 𝑃 = (Base‘𝐺)
istrkg.d = (dist‘𝐺)
istrkg.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
istrkgld ((𝐺𝑉𝑁 ∈ (ℤ‘2)) → (𝐺DimTarskiG𝑁 ↔ ∃𝑓(𝑓:(1..^𝑁)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
Distinct variable groups:   𝑓,𝐺   𝑓,𝑗,𝑥,𝑦,𝑧,𝐼   𝑃,𝑓,𝑗,𝑥,𝑦,𝑧   ,𝑓,𝑗,𝑥,𝑦,𝑧   𝑓,𝑁,𝑗,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑥,𝑦,𝑧,𝑗)   𝑉(𝑥,𝑦,𝑧,𝑓,𝑗)

Proof of Theorem istrkgld
Dummy variables 𝑑 𝑔 𝑖 𝑛 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istrkg.p . . 3 𝑃 = (Base‘𝐺)
2 istrkg.d . . 3 = (dist‘𝐺)
3 istrkg.i . . 3 𝐼 = (Itv‘𝐺)
4 eqidd 2611 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑓 = 𝑓)
5 eqidd 2611 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (1..^𝑛) = (1..^𝑛))
6 simp1 1054 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
76eqcomd 2616 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
84, 5, 7f1eq123d 6044 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑓:(1..^𝑛)–1-1𝑃𝑓:(1..^𝑛)–1-1𝑝))
9 simp2 1055 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
109eqcomd 2616 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
1110oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑥) = ((𝑓‘1)𝑑𝑥))
1210oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑥) = ((𝑓𝑗)𝑑𝑥))
1311, 12eqeq12d 2625 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ↔ ((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥)))
1410oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑦) = ((𝑓‘1)𝑑𝑦))
1510oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑦) = ((𝑓𝑗)𝑑𝑦))
1614, 15eqeq12d 2625 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ↔ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦)))
1710oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑧) = ((𝑓‘1)𝑑𝑧))
1810oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑧) = ((𝑓𝑗)𝑑𝑧))
1917, 18eqeq12d 2625 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧) ↔ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)))
2013, 16, 193anbi123d 1391 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ↔ (((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧))))
2120ralbidv 2969 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ↔ ∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧))))
22 simp3 1056 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
2322eqcomd 2616 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
2423oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
2524eleq2d 2673 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
2623oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧𝐼𝑦) = (𝑧𝑖𝑦))
2726eleq2d 2673 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧𝑖𝑦)))
2823oveqd 6566 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑧) = (𝑥𝑖𝑧))
2928eleq2d 2673 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥𝑖𝑧)))
3025, 27, 293orbi123d 1390 . . . . . . . . . 10 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
3130notbid 307 . . . . . . . . 9 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))
3221, 31anbi12d 743 . . . . . . . 8 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
337, 32rexeqbidv 3130 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
347, 33rexeqbidv 3130 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
357, 34rexeqbidv 3130 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))))
368, 35anbi12d 743 . . . 4 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓:(1..^𝑛)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))))
3736exbidv 1837 . . 3 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (∃𝑓(𝑓:(1..^𝑛)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ∃𝑓(𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))))
381, 2, 3, 37sbcie3s 15745 . 2 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑓(𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)))) ↔ ∃𝑓(𝑓:(1..^𝑛)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
39 eqidd 2611 . . . . 5 (𝑛 = 𝑁𝑓 = 𝑓)
40 oveq2 6557 . . . . 5 (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
41 eqidd 2611 . . . . 5 (𝑛 = 𝑁𝑃 = 𝑃)
4239, 40, 41f1eq123d 6044 . . . 4 (𝑛 = 𝑁 → (𝑓:(1..^𝑛)–1-1𝑃𝑓:(1..^𝑁)–1-1𝑃))
43 oveq2 6557 . . . . . . . 8 (𝑛 = 𝑁 → (2..^𝑛) = (2..^𝑁))
4443raleqdv 3121 . . . . . . 7 (𝑛 = 𝑁 → (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ↔ ∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧))))
4544anbi1d 737 . . . . . 6 (𝑛 = 𝑁 → ((∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4645rexbidv 3034 . . . . 5 (𝑛 = 𝑁 → (∃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
47462rexbidv 3039 . . . 4 (𝑛 = 𝑁 → (∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))) ↔ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))))
4842, 47anbi12d 743 . . 3 (𝑛 = 𝑁 → ((𝑓:(1..^𝑛)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ (𝑓:(1..^𝑁)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
4948exbidv 1837 . 2 (𝑛 = 𝑁 → (∃𝑓(𝑓:(1..^𝑛)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)))) ↔ ∃𝑓(𝑓:(1..^𝑁)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
50 df-trkgld 25151 . 2 DimTarskiG≥ = {⟨𝑔, 𝑛⟩ ∣ [(Base‘𝑔) / 𝑝][(dist‘𝑔) / 𝑑][(Itv‘𝑔) / 𝑖]𝑓(𝑓:(1..^𝑛)–1-1𝑝 ∧ ∃𝑥𝑝𝑦𝑝𝑧𝑝 (∀𝑗 ∈ (2..^𝑛)(((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥) ∧ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦) ∧ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))))}
5138, 49, 50brabg 4919 1 ((𝐺𝑉𝑁 ∈ (ℤ‘2)) → (𝐺DimTarskiG𝑁 ↔ ∃𝑓(𝑓:(1..^𝑁)–1-1𝑃 ∧ ∃𝑥𝑃𝑦𝑃𝑧𝑃 (∀𝑗 ∈ (2..^𝑁)(((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ∧ ((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ∧ ((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧)) ∧ ¬ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  [wsbc 3402   class class class wbr 4583  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  1c1 9816  2c2 10947  ℤ≥cuz 11563  ..^cfzo 12334  Basecbs 15695  distcds 15777  DimTarskiG≥cstrkgld 25133  Itvcitv 25135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-ov 6552  df-trkgld 25151 This theorem is referenced by:  istrkg2ld  25159  istrkg3ld  25160
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