Proof of Theorem opphllem4
| Step | Hyp | Ref
| Expression |
| 1 | | hpg.p |
. 2
⊢ 𝑃 = (Base‘𝐺) |
| 2 | | hpg.d |
. 2
⊢ − =
(dist‘𝐺) |
| 3 | | hpg.i |
. 2
⊢ 𝐼 = (Itv‘𝐺) |
| 4 | | hpg.o |
. 2
⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
| 5 | | opphl.l |
. 2
⊢ 𝐿 = (LineG‘𝐺) |
| 6 | | opphl.d |
. 2
⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
| 7 | | opphl.g |
. 2
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| 8 | | opphllem4.u |
. 2
⊢ (𝜑 → 𝑉 ∈ 𝑃) |
| 9 | | opphllem3.u |
. 2
⊢ (𝜑 → 𝑈 ∈ 𝑃) |
| 10 | | opphllem5.n |
. . 3
⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) |
| 11 | | eqid 2737 |
. . . 4
⊢
(pInvG‘𝐺) =
(pInvG‘𝐺) |
| 12 | | opphllem5.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ 𝑃) |
| 13 | 1, 2, 3, 5, 11, 7,
12, 10, 9 | mircl 28669 |
. . 3
⊢ (𝜑 → (𝑁‘𝑈) ∈ 𝑃) |
| 14 | | opphllem5.s |
. . 3
⊢ (𝜑 → 𝑆 ∈ 𝐷) |
| 15 | 1, 5, 3, 7, 6, 14 | tglnpt 28557 |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ 𝑃) |
| 16 | | opphllem5.r |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ 𝐷) |
| 17 | 1, 5, 3, 7, 6, 16 | tglnpt 28557 |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ 𝑃) |
| 18 | | opphllem3.t |
. . . . . . 7
⊢ (𝜑 → 𝑅 ≠ 𝑆) |
| 19 | 18 | necomd 2996 |
. . . . . 6
⊢ (𝜑 → 𝑆 ≠ 𝑅) |
| 20 | 1, 2, 3, 5, 11, 7,
12, 10, 17 | mirbtwn 28666 |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑅)𝐼𝑅)) |
| 21 | | opphllem3.v |
. . . . . . . 8
⊢ (𝜑 → (𝑁‘𝑅) = 𝑆) |
| 22 | 21 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝑁‘𝑅)𝐼𝑅) = (𝑆𝐼𝑅)) |
| 23 | 20, 22 | eleqtrd 2843 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ (𝑆𝐼𝑅)) |
| 24 | 1, 3, 5, 7, 15, 17, 12, 19, 23 | btwnlng1 28627 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ (𝑆𝐿𝑅)) |
| 25 | 1, 3, 5, 7, 15, 17, 19, 19, 6, 14, 16 | tglinethru 28644 |
. . . . 5
⊢ (𝜑 → 𝐷 = (𝑆𝐿𝑅)) |
| 26 | 24, 25 | eleqtrrd 2844 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ 𝐷) |
| 27 | | opphllem5.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 28 | | opphllem5.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| 29 | | opphllem5.o |
. . . . . . 7
⊢ (𝜑 → 𝐴𝑂𝐶) |
| 30 | 1, 2, 3, 4, 5, 6, 7, 27, 28, 29 | oppne1 28749 |
. . . . . 6
⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
| 31 | | opphl.k |
. . . . . . . . . . 11
⊢ 𝐾 = (hlG‘𝐺) |
| 32 | | opphllem4.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈(𝐾‘𝑅)𝐴) |
| 33 | 1, 3, 31, 9, 27, 17, 7, 32 | hlne1 28613 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 ≠ 𝑅) |
| 34 | 33 | necomd 2996 |
. . . . . . . . 9
⊢ (𝜑 → 𝑅 ≠ 𝑈) |
| 35 | 1, 3, 31, 9, 27, 17, 7, 5, 32 | hlln 28615 |
. . . . . . . . 9
⊢ (𝜑 → 𝑈 ∈ (𝐴𝐿𝑅)) |
| 36 | 1, 3, 31, 9, 27, 17, 7, 32 | hlne2 28614 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ≠ 𝑅) |
| 37 | 1, 3, 5, 7, 17, 9,
27, 34, 35, 36 | lnrot1 28631 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈ (𝑅𝐿𝑈)) |
| 38 | 37 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ (𝑅𝐿𝑈)) |
| 39 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐺 ∈ TarskiG) |
| 40 | 17 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝑃) |
| 41 | 9 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝑃) |
| 42 | 34 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ≠ 𝑈) |
| 43 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 ∈ ran 𝐿) |
| 44 | 16 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑅 ∈ 𝐷) |
| 45 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝑈 ∈ 𝐷) |
| 46 | 1, 3, 5, 39, 40, 41, 42, 42, 43, 44, 45 | tglinethru 28644 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐷 = (𝑅𝐿𝑈)) |
| 47 | 38, 46 | eleqtrrd 2844 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑈 ∈ 𝐷) → 𝐴 ∈ 𝐷) |
| 48 | 30, 47 | mtand 816 |
. . . . 5
⊢ (𝜑 → ¬ 𝑈 ∈ 𝐷) |
| 49 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐺 ∈ TarskiG) |
| 50 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝑃) |
| 51 | 9 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝑃) |
| 52 | 1, 2, 3, 5, 11, 49, 50, 10, 51 | mirmir 28670 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) = 𝑈) |
| 53 | 6 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝐷 ∈ ran 𝐿) |
| 54 | 26 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑀 ∈ 𝐷) |
| 55 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘𝑈) ∈ 𝐷) |
| 56 | 1, 2, 3, 5, 11, 49, 10, 53, 54, 55 | mirln 28684 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → (𝑁‘(𝑁‘𝑈)) ∈ 𝐷) |
| 57 | 52, 56 | eqeltrrd 2842 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁‘𝑈) ∈ 𝐷) → 𝑈 ∈ 𝐷) |
| 58 | 48, 57 | mtand 816 |
. . . 4
⊢ (𝜑 → ¬ (𝑁‘𝑈) ∈ 𝐷) |
| 59 | 1, 2, 3, 5, 11, 7,
12, 10, 9 | mirbtwn 28666 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ((𝑁‘𝑈)𝐼𝑈)) |
| 60 | 1, 2, 3, 4, 13, 9,
26, 58, 48, 59 | islnoppd 28748 |
. . 3
⊢ (𝜑 → (𝑁‘𝑈)𝑂𝑈) |
| 61 | | eqidd 2738 |
. . 3
⊢ (𝜑 → (𝑁‘𝑈) = (𝑁‘𝑈)) |
| 62 | | opphllem5.p |
. . . . . . . 8
⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) |
| 63 | | opphllem5.q |
. . . . . . . 8
⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) |
| 64 | | opphllem3.l |
. . . . . . . 8
⊢ (𝜑 → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) |
| 65 | 1, 2, 3, 4, 5, 6, 7, 31, 10, 27, 28, 16, 14, 12, 29, 62, 63, 18, 64, 9, 21 | opphllem3 28757 |
. . . . . . 7
⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 66 | 32, 65 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝐶) |
| 67 | | opphllem4.2 |
. . . . . . 7
⊢ (𝜑 → 𝑉(𝐾‘𝑆)𝐶) |
| 68 | 1, 3, 31, 8, 28, 15, 7, 67 | hlcomd 28612 |
. . . . . 6
⊢ (𝜑 → 𝐶(𝐾‘𝑆)𝑉) |
| 69 | 1, 3, 31, 13, 28, 8, 7, 15, 66, 68 | hltr 28618 |
. . . . 5
⊢ (𝜑 → (𝑁‘𝑈)(𝐾‘𝑆)𝑉) |
| 70 | 1, 3, 31, 13, 8, 15, 7 | ishlg 28610 |
. . . . 5
⊢ (𝜑 → ((𝑁‘𝑈)(𝐾‘𝑆)𝑉 ↔ ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))))) |
| 71 | 69, 70 | mpbid 232 |
. . . 4
⊢ (𝜑 → ((𝑁‘𝑈) ≠ 𝑆 ∧ 𝑉 ≠ 𝑆 ∧ ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈))))) |
| 72 | 71 | simp1d 1143 |
. . 3
⊢ (𝜑 → (𝑁‘𝑈) ≠ 𝑆) |
| 73 | 1, 3, 31, 28, 8, 15, 7, 68 | hlne2 28614 |
. . 3
⊢ (𝜑 → 𝑉 ≠ 𝑆) |
| 74 | 71 | simp3d 1145 |
. . 3
⊢ (𝜑 → ((𝑁‘𝑈) ∈ (𝑆𝐼𝑉) ∨ 𝑉 ∈ (𝑆𝐼(𝑁‘𝑈)))) |
| 75 | 1, 2, 3, 4, 5, 6, 7, 10, 13, 8, 9, 14, 60, 26, 61, 72, 73, 74 | opphllem2 28756 |
. 2
⊢ (𝜑 → 𝑉𝑂𝑈) |
| 76 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 75 | oppcom 28752 |
1
⊢ (𝜑 → 𝑈𝑂𝑉) |