| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2737 | . . . . 5
⊢
[〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear | 
| 2 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁)) | 
| 3 | 2 | eleq2d 2827 | . . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁))) | 
| 4 | 2 | eleq2d 2827 | . . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁))) | 
| 5 | 3, 4 | 3anbi12d 1439 | . . . . . . 7
⊢ (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵))) | 
| 6 | 5 | anbi1d 631 | . . . . . 6
⊢ (𝑛 = 𝑁 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 7 | 6 | rspcev 3622 | . . . . 5
⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear )) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear )) | 
| 8 | 1, 7 | mpanr2 704 | . . . 4
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear )) | 
| 9 |  | simpr1 1195 | . . . . 5
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ (𝔼‘𝑁)) | 
| 10 |  | simpr2 1196 | . . . . 5
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ (𝔼‘𝑁)) | 
| 11 |  | colinearex 36061 | . . . . . . . 8
⊢  Colinear
∈ V | 
| 12 | 11 | cnvex 7947 | . . . . . . 7
⊢ ◡ Colinear ∈ V | 
| 13 |  | ecexg 8749 | . . . . . . 7
⊢ (◡ Colinear ∈ V → [〈𝐴, 𝐵〉]◡ Colinear ∈ V) | 
| 14 | 12, 13 | ax-mp 5 | . . . . . 6
⊢
[〈𝐴, 𝐵〉]◡ Colinear ∈ V | 
| 15 |  | eleq1 2829 | . . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛))) | 
| 16 |  | neeq1 3003 | . . . . . . . . . 10
⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏)) | 
| 17 | 15, 16 | 3anbi13d 1440 | . . . . . . . . 9
⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏))) | 
| 18 |  | opeq1 4873 | . . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | 
| 19 | 18 | eceq1d 8785 | . . . . . . . . . 10
⊢ (𝑎 = 𝐴 → [〈𝑎, 𝑏〉]◡ Colinear = [〈𝐴, 𝑏〉]◡ Colinear ) | 
| 20 | 19 | eqeq2d 2748 | . . . . . . . . 9
⊢ (𝑎 = 𝐴 → (𝑙 = [〈𝑎, 𝑏〉]◡ Colinear ↔ 𝑙 = [〈𝐴, 𝑏〉]◡ Colinear )) | 
| 21 | 17, 20 | anbi12d 632 | . . . . . . . 8
⊢ (𝑎 = 𝐴 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [〈𝐴, 𝑏〉]◡ Colinear ))) | 
| 22 | 21 | rexbidv 3179 | . . . . . . 7
⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [〈𝐴, 𝑏〉]◡ Colinear ))) | 
| 23 |  | eleq1 2829 | . . . . . . . . . 10
⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛))) | 
| 24 |  | neeq2 3004 | . . . . . . . . . 10
⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵)) | 
| 25 | 23, 24 | 3anbi23d 1441 | . . . . . . . . 9
⊢ (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵))) | 
| 26 |  | opeq2 4874 | . . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | 
| 27 | 26 | eceq1d 8785 | . . . . . . . . . 10
⊢ (𝑏 = 𝐵 → [〈𝐴, 𝑏〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ) | 
| 28 | 27 | eqeq2d 2748 | . . . . . . . . 9
⊢ (𝑏 = 𝐵 → (𝑙 = [〈𝐴, 𝑏〉]◡ Colinear ↔ 𝑙 = [〈𝐴, 𝐵〉]◡ Colinear )) | 
| 29 | 25, 28 | anbi12d 632 | . . . . . . . 8
⊢ (𝑏 = 𝐵 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [〈𝐴, 𝑏〉]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 30 | 29 | rexbidv 3179 | . . . . . . 7
⊢ (𝑏 = 𝐵 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [〈𝐴, 𝑏〉]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 31 |  | eqeq1 2741 | . . . . . . . . 9
⊢ (𝑙 = [〈𝐴, 𝐵〉]◡ Colinear → (𝑙 = [〈𝐴, 𝐵〉]◡ Colinear ↔ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear )) | 
| 32 | 31 | anbi2d 630 | . . . . . . . 8
⊢ (𝑙 = [〈𝐴, 𝐵〉]◡ Colinear → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [〈𝐴, 𝐵〉]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 33 | 32 | rexbidv 3179 | . . . . . . 7
⊢ (𝑙 = [〈𝐴, 𝐵〉]◡ Colinear → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [〈𝐴, 𝐵〉]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 34 | 22, 30, 33 | eloprabg 7543 | . . . . . 6
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ [〈𝐴, 𝐵〉]◡ Colinear ∈ V) →
(〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 35 | 14, 34 | mp3an3 1452 | . . . . 5
⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 36 | 9, 10, 35 | syl2anc 584 | . . . 4
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [〈𝐴, 𝐵〉]◡ Colinear = [〈𝐴, 𝐵〉]◡ Colinear ))) | 
| 37 | 8, 36 | mpbird 257 | . . 3
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )}) | 
| 38 |  | df-ov 7434 | . . . 4
⊢ (𝐴Line𝐵) = (Line‘〈𝐴, 𝐵〉) | 
| 39 |  | df-br 5144 | . . . . . 6
⊢
(〈𝐴, 𝐵〉Line[〈𝐴, 𝐵〉]◡ Colinear ↔ 〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈
Line) | 
| 40 |  | df-line2 36138 | . . . . . . 7
⊢ Line =
{〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} | 
| 41 | 40 | eleq2i 2833 | . . . . . 6
⊢
(〈〈𝐴,
𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ Line ↔
〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )}) | 
| 42 | 39, 41 | bitri 275 | . . . . 5
⊢
(〈𝐴, 𝐵〉Line[〈𝐴, 𝐵〉]◡ Colinear ↔ 〈〈𝐴, 𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )}) | 
| 43 |  | funline 36143 | . . . . . 6
⊢ Fun
Line | 
| 44 |  | funbrfv 6957 | . . . . . 6
⊢ (Fun Line
→ (〈𝐴, 𝐵〉Line[〈𝐴, 𝐵〉]◡ Colinear → (Line‘〈𝐴, 𝐵〉) = [〈𝐴, 𝐵〉]◡ Colinear )) | 
| 45 | 43, 44 | ax-mp 5 | . . . . 5
⊢
(〈𝐴, 𝐵〉Line[〈𝐴, 𝐵〉]◡ Colinear → (Line‘〈𝐴, 𝐵〉) = [〈𝐴, 𝐵〉]◡ Colinear ) | 
| 46 | 42, 45 | sylbir 235 | . . . 4
⊢
(〈〈𝐴,
𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} → (Line‘〈𝐴, 𝐵〉) = [〈𝐴, 𝐵〉]◡ Colinear ) | 
| 47 | 38, 46 | eqtrid 2789 | . . 3
⊢
(〈〈𝐴,
𝐵〉, [〈𝐴, 𝐵〉]◡ Colinear 〉 ∈ {〈〈𝑎, 𝑏〉, 𝑙〉 ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [〈𝑎, 𝑏〉]◡ Colinear )} → (𝐴Line𝐵) = [〈𝐴, 𝐵〉]◡ Colinear ) | 
| 48 | 37, 47 | syl 17 | . 2
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = [〈𝐴, 𝐵〉]◡ Colinear ) | 
| 49 |  | opex 5469 | . . . 4
⊢
〈𝐴, 𝐵〉 ∈ V | 
| 50 |  | dfec2 8748 | . . . 4
⊢
(〈𝐴, 𝐵〉 ∈ V →
[〈𝐴, 𝐵〉]◡ Colinear = {𝑥 ∣ 〈𝐴, 𝐵〉◡ Colinear 𝑥}) | 
| 51 | 49, 50 | ax-mp 5 | . . 3
⊢
[〈𝐴, 𝐵〉]◡ Colinear = {𝑥 ∣ 〈𝐴, 𝐵〉◡ Colinear 𝑥} | 
| 52 |  | vex 3484 | . . . . 5
⊢ 𝑥 ∈ V | 
| 53 | 49, 52 | brcnv 5893 | . . . 4
⊢
(〈𝐴, 𝐵〉◡ Colinear 𝑥 ↔ 𝑥 Colinear 〈𝐴, 𝐵〉) | 
| 54 | 53 | abbii 2809 | . . 3
⊢ {𝑥 ∣ 〈𝐴, 𝐵〉◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear 〈𝐴, 𝐵〉} | 
| 55 | 51, 54 | eqtri 2765 | . 2
⊢
[〈𝐴, 𝐵〉]◡ Colinear = {𝑥 ∣ 𝑥 Colinear 〈𝐴, 𝐵〉} | 
| 56 | 48, 55 | eqtrdi 2793 | 1
⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear 〈𝐴, 𝐵〉}) |