Proof of Theorem opphllem6
| Step | Hyp | Ref
| Expression |
| 1 | | hpg.p |
. . . 4
⊢ 𝑃 = (Base‘𝐺) |
| 2 | | hpg.d |
. . . 4
⊢ − =
(dist‘𝐺) |
| 3 | | hpg.i |
. . . 4
⊢ 𝐼 = (Itv‘𝐺) |
| 4 | | opphl.l |
. . . 4
⊢ 𝐿 = (LineG‘𝐺) |
| 5 | | eqid 2737 |
. . . 4
⊢
(pInvG‘𝐺) =
(pInvG‘𝐺) |
| 6 | | opphl.g |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| 7 | 6 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐺 ∈ TarskiG) |
| 8 | | opphllem5.n |
. . . 4
⊢ 𝑁 = ((pInvG‘𝐺)‘𝑀) |
| 9 | | opphl.k |
. . . 4
⊢ 𝐾 = (hlG‘𝐺) |
| 10 | | opphllem5.m |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ 𝑃) |
| 11 | 10 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ 𝑃) |
| 12 | | opphllem5.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| 13 | 12 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ∈ 𝑃) |
| 14 | | opphllem5.c |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| 15 | 14 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ∈ 𝑃) |
| 16 | | opphllem5.u |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ 𝑃) |
| 17 | 16 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑈 ∈ 𝑃) |
| 18 | | opphl.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
| 19 | | opphllem5.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ 𝐷) |
| 20 | 1, 4, 3, 6, 18, 19 | tglnpt 28557 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ 𝑃) |
| 21 | | opphllem5.p |
. . . . . . . 8
⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) |
| 22 | 4, 6, 21 | perpln2 28719 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿) |
| 23 | 1, 3, 4, 6, 12, 20, 22 | tglnne 28636 |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ 𝑅) |
| 24 | 23 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑅) |
| 25 | | opphllem6.v |
. . . . . . . 8
⊢ (𝜑 → (𝑁‘𝑅) = 𝑆) |
| 26 | 25 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑆) |
| 27 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 = 𝑆) |
| 28 | 26, 27 | eqtr4d 2780 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑁‘𝑅) = 𝑅) |
| 29 | 1, 2, 3, 4, 5, 6, 10, 8, 20 | mirinv 28674 |
. . . . . . 7
⊢ (𝜑 → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅)) |
| 30 | 29 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑅) = 𝑅 ↔ 𝑀 = 𝑅)) |
| 31 | 28, 30 | mpbid 232 |
. . . . 5
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑅) |
| 32 | 24, 31 | neeqtrrd 3015 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐴 ≠ 𝑀) |
| 33 | | opphllem5.s |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ∈ 𝐷) |
| 34 | 1, 4, 3, 6, 18, 33 | tglnpt 28557 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ 𝑃) |
| 35 | | opphllem5.q |
. . . . . . . 8
⊢ (𝜑 → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) |
| 36 | 4, 6, 35 | perpln2 28719 |
. . . . . . 7
⊢ (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿) |
| 37 | 1, 3, 4, 6, 14, 34, 36 | tglnne 28636 |
. . . . . 6
⊢ (𝜑 → 𝐶 ≠ 𝑆) |
| 38 | 37 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑆) |
| 39 | 31, 27 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 = 𝑆) |
| 40 | 38, 39 | neeqtrrd 3015 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝐶 ≠ 𝑀) |
| 41 | | simpr 484 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡) |
| 42 | 6 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐺 ∈ TarskiG) |
| 43 | 14 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐶 ∈ 𝑃) |
| 44 | 20 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝑃) |
| 45 | 6 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
| 46 | 18 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿) |
| 47 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝐷) |
| 48 | 1, 4, 3, 45, 46, 47 | tglnpt 28557 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ 𝑃) |
| 49 | 48 | adantr 480 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝑃) |
| 50 | 12 | ad4antr 732 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐴 ∈ 𝑃) |
| 51 | 34 | ad4antr 732 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑆 ∈ 𝑃) |
| 52 | | simpllr 776 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆) |
| 53 | 1, 3, 4, 6, 14, 34, 37 | tglinerflx2 28642 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ∈ (𝐶𝐿𝑆)) |
| 54 | 53 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆)) |
| 55 | 52, 54 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆)) |
| 56 | 55 | adantr 480 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐶𝐿𝑆)) |
| 57 | 1, 2, 3, 4, 6, 18,
36, 35 | perpcom 28721 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷) |
| 58 | 57 | ad4antr 732 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷) |
| 59 | | simpr 484 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ≠ 𝑡) |
| 60 | 18 | ad4antr 732 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 ∈ ran 𝐿) |
| 61 | 19 | ad4antr 732 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ 𝐷) |
| 62 | | simpllr 776 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ 𝐷) |
| 63 | 1, 3, 4, 42, 44, 49, 59, 59, 60, 61, 62 | tglinethru 28644 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝐷 = (𝑅𝐿𝑡)) |
| 64 | 58, 63 | breqtrd 5169 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡)) |
| 65 | 1, 2, 3, 4, 42, 43, 51, 56, 49, 64 | perprag 28734 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 〈“𝐶𝑅𝑡”〉 ∈ (∟G‘𝐺)) |
| 66 | 1, 3, 4, 6, 12, 20, 23 | tglinerflx2 28642 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ (𝐴𝐿𝑅)) |
| 67 | 66 | ad4antr 732 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 ∈ (𝐴𝐿𝑅)) |
| 68 | 1, 2, 3, 4, 6, 18,
22, 21 | perpcom 28721 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷) |
| 69 | 68 | ad4antr 732 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷) |
| 70 | 69, 63 | breqtrd 5169 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡)) |
| 71 | 1, 2, 3, 4, 42, 50, 44, 67, 49, 70 | perprag 28734 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 〈“𝐴𝑅𝑡”〉 ∈ (∟G‘𝐺)) |
| 72 | | simplr 769 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐴𝐼𝐶)) |
| 73 | 1, 2, 3, 42, 50, 49, 43, 72 | tgbtwncom 28496 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑡 ∈ (𝐶𝐼𝐴)) |
| 74 | 1, 2, 3, 4, 5, 42,
43, 44, 49, 50, 65, 71, 73 | ragflat2 28711 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 ≠ 𝑡) → 𝑅 = 𝑡) |
| 75 | 41, 74 | pm2.61dane 3029 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡) |
| 76 | | simpr 484 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶)) |
| 77 | 75, 76 | eqeltrd 2841 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑅 = 𝑆) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶)) |
| 78 | | opphllem5.o |
. . . . . . . . 9
⊢ (𝜑 → 𝐴𝑂𝐶) |
| 79 | | hpg.o |
. . . . . . . . . 10
⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
| 80 | 1, 2, 3, 79, 12, 14 | islnopp 28747 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)))) |
| 81 | 78, 80 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶))) |
| 82 | 81 | simprd 495 |
. . . . . . 7
⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)) |
| 83 | 82 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐶)) |
| 84 | 77, 83 | r19.29a 3162 |
. . . . 5
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶)) |
| 85 | 31, 84 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶)) |
| 86 | 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 85 | mirbtwnhl 28688 |
. . 3
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑀)𝐶)) |
| 87 | 31 | fveq2d 6910 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑅)) |
| 88 | 87 | breqd 5154 |
. . 3
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑀)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴)) |
| 89 | 39 | fveq2d 6910 |
. . . 4
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝐾‘𝑀) = (𝐾‘𝑆)) |
| 90 | 89 | breqd 5154 |
. . 3
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → ((𝑁‘𝑈)(𝐾‘𝑀)𝐶 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 91 | 86, 88, 90 | 3bitr3d 309 |
. 2
⊢ ((𝜑 ∧ 𝑅 = 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 92 | 18 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷 ∈ ran 𝐿) |
| 93 | 6 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐺 ∈ TarskiG) |
| 94 | 12 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴 ∈ 𝑃) |
| 95 | 14 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐶 ∈ 𝑃) |
| 96 | 19 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ∈ 𝐷) |
| 97 | 33 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑆 ∈ 𝐷) |
| 98 | 10 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑀 ∈ 𝑃) |
| 99 | 78 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐴𝑂𝐶) |
| 100 | 21 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) |
| 101 | 35 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) |
| 102 | | simplr 769 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑅 ≠ 𝑆) |
| 103 | | simpr 484 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) |
| 104 | 16 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → 𝑈 ∈ 𝑃) |
| 105 | 25 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑁‘𝑅) = 𝑆) |
| 106 | 1, 2, 3, 79, 4, 92, 93, 9, 8, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105 | opphllem3 28757 |
. . 3
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 107 | 18 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷 ∈ ran 𝐿) |
| 108 | 6 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐺 ∈ TarskiG) |
| 109 | 14 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶 ∈ 𝑃) |
| 110 | 12 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴 ∈ 𝑃) |
| 111 | 33 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ∈ 𝐷) |
| 112 | 19 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝐷) |
| 113 | 10 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑀 ∈ 𝑃) |
| 114 | 78 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐴𝑂𝐶) |
| 115 | 1, 2, 3, 79, 4, 107, 108, 110, 109, 114 | oppcom 28752 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐶𝑂𝐴) |
| 116 | 35 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆)) |
| 117 | 21 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅)) |
| 118 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑅 ≠ 𝑆) |
| 119 | 118 | necomd 2996 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → 𝑆 ≠ 𝑅) |
| 120 | 119 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑆 ≠ 𝑅) |
| 121 | | simpr 484 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) |
| 122 | 16 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑈 ∈ 𝑃) |
| 123 | 1, 2, 3, 4, 5, 108, 113, 8, 122 | mircl 28669 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑈) ∈ 𝑃) |
| 124 | 20 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → 𝑅 ∈ 𝑃) |
| 125 | 25 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑅) = 𝑆) |
| 126 | 1, 2, 3, 4, 5, 108, 113, 8, 124, 125 | mircom 28671 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘𝑆) = 𝑅) |
| 127 | 1, 2, 3, 79, 4, 107, 108, 9, 8, 109, 110, 111, 112, 113, 115, 116, 117, 120, 121, 123, 126 | opphllem3 28757 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘𝑈)(𝐾‘𝑆)𝐶 ↔ (𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴)) |
| 128 | 1, 2, 3, 4, 5, 108, 113, 8, 122 | mirmir 28670 |
. . . . 5
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑁‘(𝑁‘𝑈)) = 𝑈) |
| 129 | 128 | breq1d 5153 |
. . . 4
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → ((𝑁‘(𝑁‘𝑈))(𝐾‘𝑅)𝐴 ↔ 𝑈(𝐾‘𝑅)𝐴)) |
| 130 | 127, 129 | bitr2d 280 |
. . 3
⊢ (((𝜑 ∧ 𝑅 ≠ 𝑆) ∧ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶)) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 131 | | eqid 2737 |
. . . . 5
⊢
(≤G‘𝐺) =
(≤G‘𝐺) |
| 132 | 1, 2, 3, 131, 6, 34, 14, 20, 12 | legtrid 28599 |
. . . 4
⊢ (𝜑 → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))) |
| 133 | 132 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → ((𝑆 − 𝐶)(≤G‘𝐺)(𝑅 − 𝐴) ∨ (𝑅 − 𝐴)(≤G‘𝐺)(𝑆 − 𝐶))) |
| 134 | 106, 130,
133 | mpjaodan 961 |
. 2
⊢ ((𝜑 ∧ 𝑅 ≠ 𝑆) → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |
| 135 | 91, 134 | pm2.61dane 3029 |
1
⊢ (𝜑 → (𝑈(𝐾‘𝑅)𝐴 ↔ (𝑁‘𝑈)(𝐾‘𝑆)𝐶)) |