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

Theorem istrkgld 25579
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 2772 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑓 = 𝑓)
5 eqidd 2772 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (1..^𝑛) = (1..^𝑛))
6 simp1 1130 . . . . . . 7 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑝 = 𝑃)
76eqcomd 2777 . . . . . 6 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑃 = 𝑝)
84, 5, 7f1eq123d 6272 . . . . 5 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑓:(1..^𝑛)–1-1𝑃𝑓:(1..^𝑛)–1-1𝑝))
9 simp2 1131 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑑 = )
109eqcomd 2777 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → = 𝑑)
1110oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑥) = ((𝑓‘1)𝑑𝑥))
1210oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑥) = ((𝑓𝑗)𝑑𝑥))
1311, 12eqeq12d 2786 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑥) = ((𝑓𝑗) 𝑥) ↔ ((𝑓‘1)𝑑𝑥) = ((𝑓𝑗)𝑑𝑥)))
1410oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑦) = ((𝑓‘1)𝑑𝑦))
1510oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑦) = ((𝑓𝑗)𝑑𝑦))
1614, 15eqeq12d 2786 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑦) = ((𝑓𝑗) 𝑦) ↔ ((𝑓‘1)𝑑𝑦) = ((𝑓𝑗)𝑑𝑦)))
1710oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓‘1) 𝑧) = ((𝑓‘1)𝑑𝑧))
1810oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → ((𝑓𝑗) 𝑧) = ((𝑓𝑗)𝑑𝑧))
1917, 18eqeq12d 2786 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (((𝑓‘1) 𝑧) = ((𝑓𝑗) 𝑧) ↔ ((𝑓‘1)𝑑𝑧) = ((𝑓𝑗)𝑑𝑧)))
2013, 16, 193anbi123d 1547 . . . . . . . . . 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 3135 . . . . . . . . 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 1132 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝑖 = 𝐼)
2322eqcomd 2777 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → 𝐼 = 𝑖)
2423oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑦) = (𝑥𝑖𝑦))
2524eleq2d 2836 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥𝑖𝑦)))
2623oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑧𝐼𝑦) = (𝑧𝑖𝑦))
2726eleq2d 2836 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧𝑖𝑦)))
2823oveqd 6810 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑥𝐼𝑧) = (𝑥𝑖𝑧))
2928eleq2d 2836 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑑 = 𝑖 = 𝐼) → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥𝑖𝑧)))
3025, 27, 293orbi123d 1546 . . . . . . . . . 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 616 . . . . . . . 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 3302 . . . . . . 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 3302 . . . . . 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 3302 . . . . 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 616 . . . 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 2002 . . 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 16124 . 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 2772 . . . . 5 (𝑛 = 𝑁𝑓 = 𝑓)
40 oveq2 6801 . . . . 5 (𝑛 = 𝑁 → (1..^𝑛) = (1..^𝑁))
41 eqidd 2772 . . . . 5 (𝑛 = 𝑁𝑃 = 𝑃)
4239, 40, 41f1eq123d 6272 . . . 4 (𝑛 = 𝑁 → (𝑓:(1..^𝑛)–1-1𝑃𝑓:(1..^𝑁)–1-1𝑃))
43 oveq2 6801 . . . . . . . 8 (𝑛 = 𝑁 → (2..^𝑛) = (2..^𝑁))
4443raleqdv 3293 . . . . . . 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 615 . . . . . 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 3200 . . . . 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 3205 . . . 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 616 . . 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 2002 . 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 25572 . 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 5127 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 196  wa 382  w3o 1070  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  [wsbc 3587   class class class wbr 4786  1-1wf1 6028  cfv 6031  (class class class)co 6793  1c1 10139  2c2 11272  cuz 11888  ..^cfzo 12673  Basecbs 16064  distcds 16158  DimTarskiGcstrkgld 25554  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fv 6039  df-ov 6796  df-trkgld 25572
This theorem is referenced by:  istrkg2ld  25580  istrkg3ld  25581
  Copyright terms: Public domain W3C validator