Proof of Theorem opphllem2
| Step | Hyp | Ref
| Expression |
| 1 | | hpg.p |
. . 3
⊢ 𝑃 = (Base‘𝐺) |
| 2 | | hpg.d |
. . 3
⊢ − =
(dist‘𝐺) |
| 3 | | hpg.i |
. . 3
⊢ 𝐼 = (Itv‘𝐺) |
| 4 | | hpg.o |
. . 3
⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
| 5 | | opphl.l |
. . 3
⊢ 𝐿 = (LineG‘𝐺) |
| 6 | | opphl.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
| 7 | 6 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐷 ∈ ran 𝐿) |
| 8 | | opphl.g |
. . . 4
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| 9 | 8 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐺 ∈ TarskiG) |
| 10 | | opphllem1.c |
. . . 4
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| 11 | 10 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ∈ 𝑃) |
| 12 | | opphllem1.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| 13 | 12 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵 ∈ 𝑃) |
| 14 | | opphllem1.s |
. . . 4
⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀) |
| 15 | | eqid 2737 |
. . . . 5
⊢
(pInvG‘𝐺) =
(pInvG‘𝐺) |
| 16 | | opphllem1.m |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ 𝐷) |
| 17 | 1, 5, 3, 8, 6, 16 | tglnpt 28557 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ 𝑃) |
| 18 | 17 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝑃) |
| 19 | 1, 2, 3, 5, 15, 9,
18, 14, 13 | mircl 28669 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ∈ 𝑃) |
| 20 | 16 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝐷) |
| 21 | | opphllem1.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ 𝐷) |
| 22 | 21 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝐷) |
| 23 | 1, 2, 3, 5, 15, 9,
14, 7, 20, 22 | mirln 28684 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ∈ 𝐷) |
| 24 | | simpr 484 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) |
| 25 | | simplr 769 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷) |
| 26 | 24, 25 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) |
| 27 | 8 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG) |
| 28 | 12 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃) |
| 29 | 1, 5, 3, 8, 6, 21 | tglnpt 28557 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ 𝑃) |
| 30 | 29 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝑃) |
| 31 | | opphllem1.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 32 | 31 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃) |
| 33 | | opphllem1.y |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ≠ 𝑅) |
| 34 | 33 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝑅) |
| 35 | 34 | necomd 2996 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ≠ 𝐵) |
| 36 | | simpllr 776 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐼𝐵)) |
| 37 | 1, 3, 5, 27, 30, 28, 32, 35, 36 | btwnlng1 28627 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐿𝐵)) |
| 38 | 1, 3, 5, 27, 28, 30, 32, 34, 37 | lncom 28630 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝐵𝐿𝑅)) |
| 39 | 6 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿) |
| 40 | | simplr 769 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷) |
| 41 | 21 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝐷) |
| 42 | 1, 3, 5, 27, 28, 30, 34, 34, 39, 40, 41 | tglinethru 28644 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐵𝐿𝑅)) |
| 43 | 38, 42 | eleqtrrd 2844 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷) |
| 44 | 26, 43 | pm2.61dane 3029 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → 𝐴 ∈ 𝐷) |
| 45 | | opphllem1.o |
. . . . . . . . 9
⊢ (𝜑 → 𝐴𝑂𝐶) |
| 46 | 1, 2, 3, 4, 5, 6, 8, 31, 10, 45 | oppne1 28749 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
| 47 | 46 | ad2antrr 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷) |
| 48 | 44, 47 | pm2.65da 817 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐵 ∈ 𝐷) |
| 49 | 9 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐺 ∈ TarskiG) |
| 50 | 18 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝑃) |
| 51 | 13 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝑃) |
| 52 | 1, 2, 3, 5, 15, 49, 50, 14, 51 | mirmir 28670 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) = 𝐵) |
| 53 | 7 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐷 ∈ ran 𝐿) |
| 54 | 20 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝐷) |
| 55 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝐵) ∈ 𝐷) |
| 56 | 1, 2, 3, 5, 15, 49, 14, 53, 54, 55 | mirln 28684 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) ∈ 𝐷) |
| 57 | 52, 56 | eqeltrrd 2842 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝐷) |
| 58 | 48, 57 | mtand 816 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ (𝑆‘𝐵) ∈ 𝐷) |
| 59 | 1, 2, 3, 5, 15, 9,
18, 14, 13 | mirbtwn 28666 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ ((𝑆‘𝐵)𝐼𝐵)) |
| 60 | 1, 2, 3, 4, 19, 13, 20, 58, 48, 59 | islnoppd 28748 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵)𝑂𝐵) |
| 61 | | eqidd 2738 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) = (𝑆‘𝐵)) |
| 62 | | nelne2 3040 |
. . . . . 6
⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝑅) ≠ (𝑆‘𝐵)) |
| 63 | 23, 58, 62 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ (𝑆‘𝐵)) |
| 64 | 63 | necomd 2996 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ≠ (𝑆‘𝑅)) |
| 65 | 1, 2, 3, 4, 5, 6, 8, 31, 10, 45 | oppne2 28750 |
. . . . . . 7
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷) |
| 66 | 65 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐶 ∈ 𝐷) |
| 67 | | nelne2 3040 |
. . . . . 6
⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) → (𝑆‘𝑅) ≠ 𝐶) |
| 68 | 23, 66, 67 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ 𝐶) |
| 69 | 68 | necomd 2996 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ≠ (𝑆‘𝑅)) |
| 70 | | opphllem1.n |
. . . . . . . 8
⊢ (𝜑 → 𝐴 = (𝑆‘𝐶)) |
| 71 | 70 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (𝑆‘𝐶) = 𝐴) |
| 72 | 1, 2, 3, 5, 15, 8,
17, 14, 10, 71 | mircom 28671 |
. . . . . 6
⊢ (𝜑 → (𝑆‘𝐴) = 𝐶) |
| 73 | 72 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) = 𝐶) |
| 74 | 29 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝑃) |
| 75 | 31 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ 𝑃) |
| 76 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ (𝑅𝐼𝐵)) |
| 77 | 1, 2, 3, 5, 15, 9,
18, 14, 74, 75, 13, 76 | mirbtwni 28679 |
. . . . 5
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵))) |
| 78 | 73, 77 | eqeltrrd 2842 |
. . . 4
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵))) |
| 79 | 1, 2, 3, 4, 5, 7, 9, 14, 19, 11, 13, 23, 60, 20, 61, 64, 69, 78 | opphllem1 28755 |
. . 3
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶𝑂𝐵) |
| 80 | 1, 2, 3, 4, 5, 7, 9, 11, 13, 79 | oppcom 28752 |
. 2
⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵𝑂𝐶) |
| 81 | 6 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐷 ∈ ran 𝐿) |
| 82 | 8 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐺 ∈ TarskiG) |
| 83 | 31 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ∈ 𝑃) |
| 84 | 12 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ∈ 𝑃) |
| 85 | 10 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐶 ∈ 𝑃) |
| 86 | 21 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑅 ∈ 𝐷) |
| 87 | 45 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴𝑂𝐶) |
| 88 | 16 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑀 ∈ 𝐷) |
| 89 | 70 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 = (𝑆‘𝐶)) |
| 90 | | opphllem1.x |
. . . 4
⊢ (𝜑 → 𝐴 ≠ 𝑅) |
| 91 | 90 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ≠ 𝑅) |
| 92 | 33 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ≠ 𝑅) |
| 93 | | simpr 484 |
. . 3
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ∈ (𝑅𝐼𝐴)) |
| 94 | 1, 2, 3, 4, 5, 81,
82, 14, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93 | opphllem1 28755 |
. 2
⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵𝑂𝐶) |
| 95 | | opphllem2.z |
. 2
⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴))) |
| 96 | 80, 94, 95 | mpjaodan 961 |
1
⊢ (𝜑 → 𝐵𝑂𝐶) |