Proof of Theorem hlbtwn
| Step | Hyp | Ref
| Expression |
| 1 | | hlbtwn.2 |
. . . 4
⊢ (𝜑 → 𝐵 ≠ 𝐶) |
| 2 | | hlbtwn.3 |
. . . 4
⊢ (𝜑 → 𝐷 ≠ 𝐶) |
| 3 | 1, 2 | 2thd 265 |
. . 3
⊢ (𝜑 → (𝐵 ≠ 𝐶 ↔ 𝐷 ≠ 𝐶)) |
| 4 | | ishlg.p |
. . . . . 6
⊢ 𝑃 = (Base‘𝐺) |
| 5 | | ishlg.i |
. . . . . 6
⊢ 𝐼 = (Itv‘𝐺) |
| 6 | | hlln.1 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| 7 | 6 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG) |
| 8 | | ishlg.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| 9 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃) |
| 10 | | ishlg.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 11 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃) |
| 12 | | hltr.d |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ 𝑃) |
| 13 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐷 ∈ 𝑃) |
| 14 | | ishlg.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| 15 | 14 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃) |
| 16 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵)) |
| 17 | | hlbtwn.1 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 18 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 19 | 4, 5, 7, 9, 11, 13, 15, 16, 18 | tgbtwnconn3 28585 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))) |
| 20 | | eqid 2737 |
. . . . . . 7
⊢
(dist‘𝐺) =
(dist‘𝐺) |
| 21 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
| 22 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
| 23 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ 𝑃) |
| 24 | 14 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
| 25 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
| 26 | 17 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 27 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴)) |
| 28 | 4, 20, 5, 21, 22, 23, 24, 25, 26, 27 | tgbtwnexch 28506 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
| 29 | 28 | olcd 875 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 ∈ (𝐶𝐼𝐴)) → (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))) |
| 30 | 19, 29 | jaodan 960 |
. . . 4
⊢ ((𝜑 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) → (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))) |
| 31 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐺 ∈ TarskiG) |
| 32 | 8 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐶 ∈ 𝑃) |
| 33 | 10 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐴 ∈ 𝑃) |
| 34 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐷 ∈ 𝑃) |
| 35 | 14 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐵 ∈ 𝑃) |
| 36 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐴 ∈ (𝐶𝐼𝐷)) |
| 37 | 17 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 38 | 4, 20, 5, 31, 32, 33, 34, 35, 36, 37 | tgbtwnexch 28506 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → 𝐴 ∈ (𝐶𝐼𝐵)) |
| 39 | 38 | orcd 874 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐷)) → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
| 40 | 6 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG) |
| 41 | 8 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐶 ∈ 𝑃) |
| 42 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ 𝑃) |
| 43 | 10 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐴 ∈ 𝑃) |
| 44 | 14 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ 𝑃) |
| 45 | 2 | necomd 2996 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ≠ 𝐷) |
| 46 | 45 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐶 ≠ 𝐷) |
| 47 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ (𝐶𝐼𝐴)) |
| 48 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
| 49 | 4, 5, 40, 41, 42, 43, 44, 46, 47, 48 | tgbtwnconn1 28583 |
. . . . 5
⊢ ((𝜑 ∧ 𝐷 ∈ (𝐶𝐼𝐴)) → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
| 50 | 39, 49 | jaodan 960 |
. . . 4
⊢ ((𝜑 ∧ (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))) → (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) |
| 51 | 30, 50 | impbida 801 |
. . 3
⊢ (𝜑 → ((𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴)) ↔ (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴)))) |
| 52 | 3, 51 | 3anbi23d 1441 |
. 2
⊢ (𝜑 → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))) ↔ (𝐴 ≠ 𝐶 ∧ 𝐷 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))))) |
| 53 | | ishlg.k |
. . 3
⊢ 𝐾 = (hlG‘𝐺) |
| 54 | 4, 5, 53, 10, 14, 8, 6 | ishlg 28610 |
. 2
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ (𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐶𝐼𝐴))))) |
| 55 | 4, 5, 53, 10, 12, 8, 6 | ishlg 28610 |
. 2
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐷 ↔ (𝐴 ≠ 𝐶 ∧ 𝐷 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐷) ∨ 𝐷 ∈ (𝐶𝐼𝐴))))) |
| 56 | 52, 54, 55 | 3bitr4d 311 |
1
⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐴(𝐾‘𝐶)𝐷)) |