| Step |
Hyp |
Ref |
Expression |
| 1 |
|
opex |
⊢ ⟨ 𝐴 , 𝐵 ⟩ ∈ V |
| 2 |
|
opex |
⊢ ⟨ 𝐶 , 𝐷 ⟩ ∈ V |
| 3 |
|
eqeq1 |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ) ) |
| 4 |
|
eqcom |
⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑎 , 𝑏 ⟩ ↔ ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) |
| 5 |
3 4
|
bitrdi |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ↔ ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) ) |
| 6 |
5
|
3anbi1d |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 7 |
6
|
rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 8 |
7
|
2rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 9 |
8
|
2rexbidv |
⊢ ( 𝑝 = ⟨ 𝐴 , 𝐵 ⟩ → ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 10 |
|
eqeq1 |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ↔ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑐 , 𝑑 ⟩ ) ) |
| 11 |
|
eqcom |
⊢ ( ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝑐 , 𝑑 ⟩ ↔ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) |
| 12 |
10 11
|
bitrdi |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ↔ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) ) |
| 13 |
12
|
3anbi2d |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 14 |
13
|
rexbidv |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 15 |
14
|
2rexbidv |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 16 |
15
|
2rexbidv |
⊢ ( 𝑞 = ⟨ 𝐶 , 𝐷 ⟩ → ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 17 |
|
df-segle |
⊢ Seg≤ = { ⟨ 𝑝 , 𝑞 ⟩ ∣ ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑝 = ⟨ 𝑎 , 𝑏 ⟩ ∧ 𝑞 = ⟨ 𝑐 , 𝑑 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) } |
| 18 |
1 2 9 16 17
|
brab |
⊢ ( ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 19 |
|
vex |
⊢ 𝑎 ∈ V |
| 20 |
|
vex |
⊢ 𝑏 ∈ V |
| 21 |
19 20
|
opth |
⊢ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ) |
| 22 |
|
vex |
⊢ 𝑐 ∈ V |
| 23 |
|
vex |
⊢ 𝑑 ∈ V |
| 24 |
22 23
|
opth |
⊢ ( ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) |
| 25 |
|
biid |
⊢ ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) |
| 26 |
21 24 25
|
3anbi123i |
⊢ ( ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 27 |
26
|
2rexbii |
⊢ ( ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 28 |
27
|
2rexbii |
⊢ ( ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 29 |
28
|
rexbii |
⊢ ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 30 |
|
simpl2l |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 31 |
30
|
ad2antrl |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 32 |
|
eleenn |
⊢ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) → 𝑁 ∈ ℕ ) |
| 33 |
31 32
|
syl |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → 𝑁 ∈ ℕ ) |
| 34 |
|
simprlr |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → 𝑛 ∈ ℕ ) |
| 35 |
|
simprll |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ) |
| 36 |
35
|
adantl |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ) |
| 37 |
|
axdimuniq |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑛 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ) ) → 𝑁 = 𝑛 ) |
| 38 |
33 31 34 36 37
|
syl22anc |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → 𝑁 = 𝑛 ) |
| 39 |
38
|
fveq2d |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → ( 𝔼 ‘ 𝑁 ) = ( 𝔼 ‘ 𝑛 ) ) |
| 40 |
39
|
rexeqdv |
⊢ ( ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) ∧ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 41 |
40
|
exbiri |
⊢ ( ( 𝑎 = 𝐴 ∧ ( 𝑏 = 𝐵 ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) ) → ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 42 |
41
|
anassrs |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 43 |
|
eleq1 |
⊢ ( 𝑎 = 𝐴 → ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ↔ 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ) ) |
| 44 |
|
eleq1 |
⊢ ( 𝑏 = 𝐵 → ( 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ↔ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ) |
| 45 |
43 44
|
bi2anan9 |
⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ↔ ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) |
| 46 |
|
eleq1 |
⊢ ( 𝑐 = 𝐶 → ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ↔ 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ) ) |
| 47 |
|
eleq1 |
⊢ ( 𝑑 = 𝐷 → ( 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ↔ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) |
| 48 |
46 47
|
bi2anan9 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ( ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ↔ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) |
| 49 |
45 48
|
bi2anan9 |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ↔ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) |
| 50 |
49
|
anbi2d |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ↔ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) ) ) |
| 51 |
|
opeq12 |
⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) |
| 52 |
51
|
breq1d |
⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) |
| 53 |
52
|
anbi2d |
⊢ ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 54 |
|
opeq12 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) |
| 55 |
54
|
breq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ↔ 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ) ) |
| 56 |
|
opeq1 |
⊢ ( 𝑐 = 𝐶 → ⟨ 𝑐 , 𝑦 ⟩ = ⟨ 𝐶 , 𝑦 ⟩ ) |
| 57 |
56
|
breq2d |
⊢ ( 𝑐 = 𝐶 → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) |
| 58 |
57
|
adantr |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) |
| 59 |
55 58
|
anbi12d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 60 |
53 59
|
sylan9bb |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 61 |
60
|
rexbidv |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 62 |
61
|
imbi1d |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ↔ ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 63 |
42 50 62
|
3imtr4d |
⊢ ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 64 |
63
|
com12 |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 65 |
64
|
expd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) → ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) ) |
| 66 |
65
|
3impd |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) ) ) → ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 67 |
66
|
expr |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ) → ( ( 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ) → ( ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 68 |
67
|
rexlimdvv |
⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) ∧ ( 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ) ) → ( ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 69 |
68
|
rexlimdvva |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑛 ∈ ℕ ) → ( ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 70 |
69
|
rexlimdva |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ( 𝑎 = 𝐴 ∧ 𝑏 = 𝐵 ) ∧ ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 71 |
29 70
|
syl5bi |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 72 |
|
simpl1 |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → 𝑁 ∈ ℕ ) |
| 73 |
|
simpl2l |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 74 |
|
simpl2r |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 75 |
|
simpl3l |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 76 |
|
simpl3r |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) |
| 77 |
|
eqidd |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) |
| 78 |
|
eqidd |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) |
| 79 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) |
| 80 |
|
opeq1 |
⊢ ( 𝑐 = 𝐶 → ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝑑 ⟩ ) |
| 81 |
80
|
eqeq1d |
⊢ ( 𝑐 = 𝐶 → ( ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐶 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) ) |
| 82 |
80
|
breq2d |
⊢ ( 𝑐 = 𝐶 → ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ↔ 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ) ) |
| 83 |
82 57
|
anbi12d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 84 |
83
|
rexbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 85 |
81 84
|
3anbi23d |
⊢ ( 𝑐 = 𝐶 → ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 86 |
|
opeq2 |
⊢ ( 𝑑 = 𝐷 → ⟨ 𝐶 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) |
| 87 |
86
|
eqeq1d |
⊢ ( 𝑑 = 𝐷 → ( ⟨ 𝐶 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ) ) |
| 88 |
86
|
breq2d |
⊢ ( 𝑑 = 𝐷 → ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ↔ 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ) ) |
| 89 |
88
|
anbi1d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 90 |
89
|
rexbidv |
⊢ ( 𝑑 = 𝐷 → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 91 |
87 90
|
3anbi23d |
⊢ ( 𝑑 = 𝐷 → ( ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) ) |
| 92 |
85 91
|
rspc2ev |
⊢ ( ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) → ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 93 |
75 76 77 78 79 92
|
syl113anc |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 94 |
|
opeq1 |
⊢ ( 𝑎 = 𝐴 → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝑏 ⟩ ) |
| 95 |
94
|
eqeq1d |
⊢ ( 𝑎 = 𝐴 → ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) ) |
| 96 |
94
|
breq1d |
⊢ ( 𝑎 = 𝐴 → ( ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ↔ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) |
| 97 |
96
|
anbi2d |
⊢ ( 𝑎 = 𝐴 → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 98 |
97
|
rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 99 |
95 98
|
3anbi13d |
⊢ ( 𝑎 = 𝐴 → ( ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 100 |
99
|
2rexbidv |
⊢ ( 𝑎 = 𝐴 → ( ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 101 |
|
opeq2 |
⊢ ( 𝑏 = 𝐵 → ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) |
| 102 |
101
|
eqeq1d |
⊢ ( 𝑏 = 𝐵 → ( ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ) ) |
| 103 |
101
|
breq1d |
⊢ ( 𝑏 = 𝐵 → ( ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) |
| 104 |
103
|
anbi2d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 105 |
104
|
rexbidv |
⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 106 |
102 105
|
3anbi13d |
⊢ ( 𝑏 = 𝐵 → ( ( ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 107 |
106
|
2rexbidv |
⊢ ( 𝑏 = 𝐵 → ( ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 108 |
100 107
|
rspc2ev |
⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) → ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 109 |
73 74 93 108
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 110 |
|
fveq2 |
⊢ ( 𝑛 = 𝑁 → ( 𝔼 ‘ 𝑛 ) = ( 𝔼 ‘ 𝑁 ) ) |
| 111 |
110
|
rexeqdv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 112 |
111
|
3anbi3d |
⊢ ( 𝑛 = 𝑁 → ( ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 113 |
110 112
|
rexeqbidv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 114 |
110 113
|
rexeqbidv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 115 |
110 114
|
rexeqbidv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 116 |
110 115
|
rexeqbidv |
⊢ ( 𝑛 = 𝑁 → ( ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 117 |
116
|
rspcev |
⊢ ( ( 𝑁 ∈ ℕ ∧ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) → ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 118 |
72 109 117
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) → ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) |
| 119 |
118
|
ex |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) → ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ) ) |
| 120 |
71 119
|
impbid |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ∃ 𝑛 ∈ ℕ ∃ 𝑎 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑏 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑐 ∈ ( 𝔼 ‘ 𝑛 ) ∃ 𝑑 ∈ ( 𝔼 ‘ 𝑛 ) ( ⟨ 𝑎 , 𝑏 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝑐 , 𝑑 ⟩ = ⟨ 𝐶 , 𝐷 ⟩ ∧ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑛 ) ( 𝑦 Btwn ⟨ 𝑐 , 𝑑 ⟩ ∧ ⟨ 𝑎 , 𝑏 ⟩ Cgr ⟨ 𝑐 , 𝑦 ⟩ ) ) ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |
| 121 |
18 120
|
syl5bb |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Seg≤ ⟨ 𝐶 , 𝐷 ⟩ ↔ ∃ 𝑦 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑦 Btwn ⟨ 𝐶 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝑦 ⟩ ) ) ) |