Metamath Proof Explorer


Theorem istrkg3ld

Description: Property of fulfilling the lower dimension 3 axiom. (Contributed by Thierry Arnoux, 12-Jul-2020)

Ref Expression
Hypotheses istrkg.p 𝑃 = ( Base ‘ 𝐺 )
istrkg.d = ( dist ‘ 𝐺 )
istrkg.i 𝐼 = ( Itv ‘ 𝐺 )
Assertion istrkg3ld ( 𝐺𝑉 → ( 𝐺 DimTarskiG≥ 3 ↔ ∃ 𝑢𝑃𝑣𝑃 ( 𝑢𝑣 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )

Proof

Step Hyp Ref Expression
1 istrkg.p 𝑃 = ( Base ‘ 𝐺 )
2 istrkg.d = ( dist ‘ 𝐺 )
3 istrkg.i 𝐼 = ( Itv ‘ 𝐺 )
4 3z 3 ∈ ℤ
5 2re 2 ∈ ℝ
6 3re 3 ∈ ℝ
7 2lt3 2 < 3
8 5 6 7 ltleii 2 ≤ 3
9 2z 2 ∈ ℤ
10 9 eluz1i ( 3 ∈ ( ℤ ‘ 2 ) ↔ ( 3 ∈ ℤ ∧ 2 ≤ 3 ) )
11 4 8 10 mpbir2an 3 ∈ ( ℤ ‘ 2 )
12 1 2 3 istrkgld ( ( 𝐺𝑉 ∧ 3 ∈ ( ℤ ‘ 2 ) ) → ( 𝐺 DimTarskiG≥ 3 ↔ ∃ 𝑓 ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )
13 11 12 mpan2 ( 𝐺𝑉 → ( 𝐺 DimTarskiG≥ 3 ↔ ∃ 𝑓 ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )
14 fzo13pr ( 1 ..^ 3 ) = { 1 , 2 }
15 f1eq2 ( ( 1 ..^ 3 ) = { 1 , 2 } → ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃𝑓 : { 1 , 2 } –1-1𝑃 ) )
16 14 15 ax-mp ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃𝑓 : { 1 , 2 } –1-1𝑃 )
17 16 anbi1i ( ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
18 17 exbii ( ∃ 𝑓 ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ∃ 𝑓 ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
19 18 a1i ( 𝐺𝑉 → ( ∃ 𝑓 ( 𝑓 : ( 1 ..^ 3 ) –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ∃ 𝑓 ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )
20 1z 1 ∈ ℤ
21 1ne2 1 ≠ 2
22 oveq1 ( 𝑢 = ( 𝑓 ‘ 1 ) → ( 𝑢 𝑥 ) = ( ( 𝑓 ‘ 1 ) 𝑥 ) )
23 22 eqeq1d ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ) )
24 oveq1 ( 𝑢 = ( 𝑓 ‘ 1 ) → ( 𝑢 𝑦 ) = ( ( 𝑓 ‘ 1 ) 𝑦 ) )
25 24 eqeq1d ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ) )
26 oveq1 ( 𝑢 = ( 𝑓 ‘ 1 ) → ( 𝑢 𝑧 ) = ( ( 𝑓 ‘ 1 ) 𝑧 ) )
27 26 eqeq1d ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) )
28 23 25 27 3anbi123d ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ) )
29 28 anbi1d ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
30 29 rexbidv ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ∃ 𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ∃ 𝑧𝑃 ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
31 30 2rexbidv ( 𝑢 = ( 𝑓 ‘ 1 ) → ( ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
32 oveq1 ( 𝑣 = ( 𝑓 ‘ 2 ) → ( 𝑣 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) )
33 32 eqeq2d ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ) )
34 oveq1 ( 𝑣 = ( 𝑓 ‘ 2 ) → ( 𝑣 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) )
35 34 eqeq2d ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ) )
36 oveq1 ( 𝑣 = ( 𝑓 ‘ 2 ) → ( 𝑣 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) )
37 36 eqeq2d ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) )
38 33 35 37 3anbi123d ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) ) )
39 2p1e3 ( 2 + 1 ) = 3
40 39 oveq2i ( 2 ..^ ( 2 + 1 ) ) = ( 2 ..^ 3 )
41 fzosn ( 2 ∈ ℤ → ( 2 ..^ ( 2 + 1 ) ) = { 2 } )
42 9 41 ax-mp ( 2 ..^ ( 2 + 1 ) ) = { 2 }
43 40 42 eqtr3i ( 2 ..^ 3 ) = { 2 }
44 43 raleqi ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ↔ ∀ 𝑗 ∈ { 2 } ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) )
45 fveq2 ( 𝑗 = 2 → ( 𝑓𝑗 ) = ( 𝑓 ‘ 2 ) )
46 45 oveq1d ( 𝑗 = 2 → ( ( 𝑓𝑗 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) )
47 46 eqeq2d ( 𝑗 = 2 → ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ) )
48 45 oveq1d ( 𝑗 = 2 → ( ( 𝑓𝑗 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) )
49 48 eqeq2d ( 𝑗 = 2 → ( ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ) )
50 45 oveq1d ( 𝑗 = 2 → ( ( 𝑓𝑗 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) )
51 50 eqeq2d ( 𝑗 = 2 → ( ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ↔ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) )
52 47 49 51 3anbi123d ( 𝑗 = 2 → ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) ) )
53 52 ralsng ( 2 ∈ ℤ → ( ∀ 𝑗 ∈ { 2 } ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) ) )
54 9 53 ax-mp ( ∀ 𝑗 ∈ { 2 } ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) )
55 44 54 bitri ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ↔ ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓 ‘ 2 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓 ‘ 2 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓 ‘ 2 ) 𝑧 ) ) )
56 38 55 bitr4di ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ↔ ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ) )
57 56 anbi1d ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
58 57 rexbidv ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ∃ 𝑧𝑃 ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ∃ 𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
59 58 2rexbidv ( 𝑣 = ( 𝑓 ‘ 2 ) → ( ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ↔ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
60 31 59 f1prex ( ( 1 ∈ ℤ ∧ 2 ∈ ℤ ∧ 1 ≠ 2 ) → ( ∃ 𝑓 ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ∃ 𝑢𝑃𝑣𝑃 ( 𝑢𝑣 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )
61 20 9 21 60 mp3an ( ∃ 𝑓 ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ∃ 𝑢𝑃𝑣𝑃 ( 𝑢𝑣 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) )
62 61 a1i ( 𝐺𝑉 → ( ∃ 𝑓 ( 𝑓 : { 1 , 2 } –1-1𝑃 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ∀ 𝑗 ∈ ( 2 ..^ 3 ) ( ( ( 𝑓 ‘ 1 ) 𝑥 ) = ( ( 𝑓𝑗 ) 𝑥 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑦 ) = ( ( 𝑓𝑗 ) 𝑦 ) ∧ ( ( 𝑓 ‘ 1 ) 𝑧 ) = ( ( 𝑓𝑗 ) 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ↔ ∃ 𝑢𝑃𝑣𝑃 ( 𝑢𝑣 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )
63 13 19 62 3bitrd ( 𝐺𝑉 → ( 𝐺 DimTarskiG≥ 3 ↔ ∃ 𝑢𝑃𝑣𝑃 ( 𝑢𝑣 ∧ ∃ 𝑥𝑃𝑦𝑃𝑧𝑃 ( ( ( 𝑢 𝑥 ) = ( 𝑣 𝑥 ) ∧ ( 𝑢 𝑦 ) = ( 𝑣 𝑦 ) ∧ ( 𝑢 𝑧 ) = ( 𝑣 𝑧 ) ) ∧ ¬ ( 𝑧 ∈ ( 𝑥 𝐼 𝑦 ) ∨ 𝑥 ∈ ( 𝑧 𝐼 𝑦 ) ∨ 𝑦 ∈ ( 𝑥 𝐼 𝑧 ) ) ) ) ) )