Step | Hyp | Ref
| Expression |
1 | | outpasch.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
2 | 1 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ 𝑃) |
3 | | simpr 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) |
4 | 3 | eleq1d 2823 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐴 ∈ (𝐴𝐼𝐵))) |
5 | 3 | oveq2d 7283 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑅𝐼𝑥) = (𝑅𝐼𝐴)) |
6 | 5 | eleq2d 2824 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐴))) |
7 | 4, 6 | anbi12d 631 |
. . . . 5
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴)))) |
8 | | outpasch.p |
. . . . . . . 8
⊢ 𝑃 = (Base‘𝐺) |
9 | | eqid 2738 |
. . . . . . . 8
⊢
(dist‘𝐺) =
(dist‘𝐺) |
10 | | outpasch.i |
. . . . . . . 8
⊢ 𝐼 = (Itv‘𝐺) |
11 | | outpasch.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
12 | | outpasch.b |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
13 | 8, 9, 10, 11, 1, 12 | tgbtwntriv1 26862 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
14 | 13 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ (𝐴𝐼𝐵)) |
15 | 11 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐺 ∈ TarskiG) |
16 | | outpasch.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ 𝑃) |
17 | 16 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑅 ∈ 𝑃) |
18 | | outpasch.q |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ 𝑃) |
19 | 18 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ 𝑃) |
20 | | outpasch.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
21 | 20 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ 𝑃) |
22 | | simpr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐶)) |
23 | | outpasch.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝑅)) |
24 | 8, 9, 10, 11, 1, 20, 16, 23 | tgbtwncom 26859 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ (𝑅𝐼𝐴)) |
25 | 24 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
26 | 8, 9, 10, 15, 17, 19, 21, 2, 22, 25 | tgbtwnexch 26869 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐴)) |
27 | 14, 26 | jca 512 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴))) |
28 | 2, 7, 27 | rspcedvd 3562 |
. . . 4
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
29 | 28 | adantlr 712 |
. . 3
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
30 | 12 | ad2antrr 723 |
. . . 4
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ 𝑃) |
31 | | eleq1 2826 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐵 ∈ (𝐴𝐼𝐵))) |
32 | | oveq2 7275 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝑅𝐼𝑥) = (𝑅𝐼𝐵)) |
33 | 32 | eleq2d 2824 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐵))) |
34 | 31, 33 | anbi12d 631 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
35 | 34 | adantl 482 |
. . . 4
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
36 | 8, 9, 10, 11, 1, 12 | tgbtwntriv2 26858 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
37 | 36 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐵)) |
38 | 11 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐺 ∈ TarskiG) |
39 | 20 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐶 ∈ 𝑃) |
40 | 16 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ 𝑃) |
41 | 18 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ 𝑃) |
42 | 12 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐵 ∈ 𝑃) |
43 | | simpr 485 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝑄𝐼𝐶)) |
44 | 8, 9, 10, 38, 41, 40, 39, 43 | tgbtwncom 26859 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝑄)) |
45 | | outpasch.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ (𝐵𝐼𝐶)) |
46 | 8, 9, 10, 11, 12, 18, 20, 45 | tgbtwncom 26859 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ (𝐶𝐼𝐵)) |
47 | 46 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
48 | 8, 9, 10, 38, 39, 40, 41, 42, 44, 47 | tgbtwnexch3 26865 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
49 | 11 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐺 ∈ TarskiG) |
50 | 12 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐵 ∈ 𝑃) |
51 | 18 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ 𝑃) |
52 | 16 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑅 ∈ 𝑃) |
53 | 20 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ 𝑃) |
54 | | simpr 485 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 = 𝐶) |
55 | 8, 9, 10, 11, 16, 20 | tgbtwntriv2 26858 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ (𝑅𝐼𝐶)) |
56 | 55 | ad4antr 729 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝐶 ∈ (𝑅𝐼𝐶)) |
57 | 54, 56 | eqeltrd 2839 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 ∈ (𝑅𝐼𝐶)) |
58 | | simpllr 773 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → ¬ 𝑄 ∈ (𝑅𝐼𝐶)) |
59 | 57, 58 | pm2.65da 814 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → ¬ 𝑄 = 𝐶) |
60 | 59 | neqned 2950 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ≠ 𝐶) |
61 | 45 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝐶)) |
62 | | simpr 485 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ (𝑄𝐼𝑅)) |
63 | 8, 9, 10, 49, 50, 51, 53, 52, 60, 61, 62 | tgbtwnouttr 26868 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝑅)) |
64 | 8, 9, 10, 49, 50, 51, 52, 63 | tgbtwncom 26859 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
65 | | outpasch.l |
. . . . . . . . . 10
⊢ 𝐿 = (LineG‘𝐺) |
66 | 8, 65, 10, 11, 18, 20, 16 | tgcolg 26925 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
67 | 66 | biimpa 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
68 | | 3orcoma 1092 |
. . . . . . . . 9
⊢ ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
69 | | 3orass 1089 |
. . . . . . . . 9
⊢ ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
70 | 68, 69 | bitr3i 276 |
. . . . . . . 8
⊢ ((𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
71 | 67, 70 | sylib 217 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
72 | 71 | orcanai 1000 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
73 | 48, 64, 72 | mpjaodan 956 |
. . . . 5
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
74 | 37, 73 | jca 512 |
. . . 4
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))) |
75 | 30, 35, 74 | rspcedvd 3562 |
. . 3
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
76 | 29, 75 | pm2.61dan 810 |
. 2
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
77 | 12 | ad2antrr 723 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ 𝑃) |
78 | 34 | adantl 482 |
. . . 4
⊢ ((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
79 | 36 | ad2antrr 723 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐴𝐼𝐵)) |
80 | 11 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG) |
81 | 16 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ 𝑃) |
82 | 18 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ 𝑃) |
83 | 20 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ 𝑃) |
84 | | simplr 766 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
85 | | simpr 485 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐿𝑄)) |
86 | 11 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐺 ∈ TarskiG) |
87 | 16 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ 𝑃) |
88 | 18 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ 𝑃) |
89 | 20 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ∈ 𝑃) |
90 | | simpr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
91 | 8, 10, 65, 86, 87, 88, 89, 90 | ncolne1 26996 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ≠ 𝑄) |
92 | 8, 10, 65, 86, 87, 88, 91 | tglinerflx2 27005 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
93 | 92 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
94 | 8, 65, 10, 86, 88, 89, 87, 90 | ncolcom 26932 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝐶𝐿𝑄) ∨ 𝐶 = 𝑄)) |
95 | 8, 65, 10, 86, 89, 88, 87, 94 | ncolrot1 26933 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅)) |
96 | 8, 10, 65, 86, 89, 88, 87, 95 | ncolne1 26996 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ≠ 𝑄) |
97 | 96 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ≠ 𝑄) |
98 | 46 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
99 | 8, 10, 65, 80, 83, 82, 77, 97, 98 | btwnlng3 26992 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐶𝐿𝑄)) |
100 | 8, 10, 65, 80, 83, 82, 97 | tglinerflx2 27005 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐿𝑄)) |
101 | 8, 10, 65, 80, 81, 82, 83, 82, 84, 85, 93, 99, 100 | tglineinteq 27016 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 = 𝑄) |
102 | 8, 9, 10, 11, 16, 12 | tgbtwntriv2 26858 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝑅𝐼𝐵)) |
103 | 102 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐼𝐵)) |
104 | 101, 103 | eqeltrrd 2840 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
105 | 79, 104 | jca 512 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))) |
106 | 77, 78, 105 | rspcedvd 3562 |
. . 3
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
107 | | eleq1 2826 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑥 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑥 ∈ (𝑎𝐼𝑏))) |
108 | 107 | cbvrexvw 3381 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
(𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏)) |
109 | 108 | anbi2i 623 |
. . . . . . . 8
⊢ (((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))) |
110 | 109 | opabbii 5140 |
. . . . . . 7
⊢
{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))} = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))} |
111 | 11 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG) |
112 | 16 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ 𝑃) |
113 | 18 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ 𝑃) |
114 | 91 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ≠ 𝑄) |
115 | 8, 10, 65, 111, 112, 113, 114 | tgelrnln 27001 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝑅𝐿𝑄) ∈ ran 𝐿) |
116 | | eqid 2738 |
. . . . . . 7
⊢
(hlG‘𝐺) =
(hlG‘𝐺) |
117 | 20 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ 𝑃) |
118 | 1 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴 ∈ 𝑃) |
119 | 12 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ 𝑃) |
120 | 92 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
121 | 8, 65, 10, 86, 88, 89, 87, 90 | ncolrot2 26934 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄)) |
122 | | pm2.45 879 |
. . . . . . . . . 10
⊢ (¬
(𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
123 | 121, 122 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
124 | 123 | adantr 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
125 | | simpr 485 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐵 ∈ (𝑅𝐿𝑄)) |
126 | 46 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
127 | 8, 9, 10, 110, 117, 119, 120, 124, 125, 126 | islnoppd 27111 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵) |
128 | 8, 10, 65, 86, 87, 88, 91 | tglinerflx1 27004 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ (𝑅𝐿𝑄)) |
129 | 128 | adantr 481 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ (𝑅𝐿𝑄)) |
130 | 24 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
131 | 8, 10, 65, 86, 89, 87, 88, 121 | ncolne1 26996 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ≠ 𝑅) |
132 | 131 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ≠ 𝑅) |
133 | 8, 9, 10, 111, 112, 117, 118, 130, 132 | tgbtwnne 26861 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ≠ 𝐴) |
134 | 8, 10, 116, 112, 118, 117, 111, 118, 130, 133, 132 | btwnhl1 26983 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶((hlG‘𝐺)‘𝑅)𝐴) |
135 | 8, 9, 10, 110, 65, 115, 111, 116, 117, 118, 119, 127, 129, 134 | opphl 27125 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵) |
136 | 8, 9, 10, 110, 118, 119 | islnopp 27110 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵 ↔ ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵)))) |
137 | 135, 136 | mpbid 231 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵))) |
138 | 137 | simprd 496 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵)) |
139 | 111 | ad2antrr 723 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG) |
140 | 115 | ad2antrr 723 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑅𝐿𝑄) ∈ ran 𝐿) |
141 | | simplr 766 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝑅𝐿𝑄)) |
142 | 8, 65, 10, 139, 140, 141 | tglnpt 26920 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ 𝑃) |
143 | | simpr 485 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐴𝐼𝐵)) |
144 | 139 | ad2antrr 723 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG) |
145 | 87 | ad3antrrr 727 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑅 ∈ 𝑃) |
146 | 145 | ad2antrr 723 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅 ∈ 𝑃) |
147 | 88 | ad5antr 731 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ 𝑃) |
148 | 117 | ad2antrr 723 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ 𝑃) |
149 | 148 | ad2antrr 723 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶 ∈ 𝑃) |
150 | 90 | ad5antr 731 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
151 | | simplr 766 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ 𝑃) |
152 | 114 | ad4antr 729 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅 ≠ 𝑄) |
153 | 142 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ 𝑃) |
154 | 91 | necomd 2999 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ≠ 𝑅) |
155 | 154 | ad5antr 731 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ≠ 𝑅) |
156 | 141 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑅𝐿𝑄)) |
157 | 8, 10, 65, 144, 147, 146, 153, 155, 156 | lncom 26993 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑄𝐿𝑅)) |
158 | | simprl 768 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑥𝐼𝑅)) |
159 | 8, 10, 65, 144, 153, 147, 146, 151, 157, 158 | coltr3 27019 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝑅)) |
160 | 8, 10, 65, 144, 146, 147, 151, 152, 159 | lncom 26993 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐿𝑄)) |
161 | 92 | ad5antr 731 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐿𝑄)) |
162 | 96 | ad5antr 731 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶 ≠ 𝑄) |
163 | 119 | ad2antrr 723 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃) |
164 | 163 | ad2antrr 723 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ 𝑃) |
165 | 12 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ 𝑃) |
166 | 96 | necomd 2999 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ≠ 𝐶) |
167 | 45 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐵𝐼𝐶)) |
168 | 8, 10, 65, 86, 88, 89, 165, 166, 167 | btwnlng2 26991 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ (𝑄𝐿𝐶)) |
169 | 168 | ad5antr 731 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ (𝑄𝐿𝐶)) |
170 | | simprr 770 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐼𝐵)) |
171 | 8, 9, 10, 144, 149, 151, 164, 170 | tgbtwncom 26859 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐵𝐼𝐶)) |
172 | 8, 10, 65, 144, 164, 147, 149, 151, 169, 171 | coltr3 27019 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝐶)) |
173 | 8, 10, 65, 144, 149, 147, 151, 162, 172 | lncom 26993 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐿𝑄)) |
174 | 8, 10, 65, 86, 89, 88, 96 | tglinerflx2 27005 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐶𝐿𝑄)) |
175 | 174 | ad5antr 731 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝐶𝐿𝑄)) |
176 | 8, 10, 65, 144, 146, 147, 149, 147, 150, 160, 161, 173, 175 | tglineinteq 27016 |
. . . . . . . . 9
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 = 𝑄) |
177 | 8, 9, 10, 144, 153, 151, 146, 158 | tgbtwncom 26859 |
. . . . . . . . 9
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐼𝑥)) |
178 | 176, 177 | eqeltrrd 2840 |
. . . . . . . 8
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐼𝑥)) |
179 | 118 | ad2antrr 723 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃) |
180 | 8, 9, 10, 139, 179, 142, 163, 143 | tgbtwncom 26859 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐵𝐼𝐴)) |
181 | 24 | ad4antr 729 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
182 | 8, 9, 10, 139, 163, 145, 179, 142, 148, 180, 181 | axtgpasch 26838 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → ∃𝑡 ∈ 𝑃 (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) |
183 | 178, 182 | r19.29a 3216 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑄 ∈ (𝑅𝐼𝑥)) |
184 | 142, 143,
183 | jca32 516 |
. . . . . 6
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))) |
185 | 184 | expl 458 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((𝑥 ∈ (𝑅𝐿𝑄) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))))) |
186 | 185 | reximdv2 3197 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))) |
187 | 138, 186 | mpd 15 |
. . 3
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
188 | 106, 187 | pm2.61dan 810 |
. 2
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
189 | 76, 188 | pm2.61dan 810 |
1
⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |