Step | Hyp | Ref
| Expression |
1 | | outpasch.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
2 | 1 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ 𝑃) |
3 | | simpr 476 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) |
4 | 3 | eleq1d 2672 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐴 ∈ (𝐴𝐼𝐵))) |
5 | 3 | oveq2d 6565 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑅𝐼𝑥) = (𝑅𝐼𝐴)) |
6 | 5 | eleq2d 2673 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐴))) |
7 | 4, 6 | anbi12d 743 |
. . . . 5
⊢ (((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴)))) |
8 | | outpasch.p |
. . . . . . . 8
⊢ 𝑃 = (Base‘𝐺) |
9 | | eqid 2610 |
. . . . . . . 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 25186 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵)) |
14 | 13 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ (𝐴𝐼𝐵)) |
15 | 11 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐺 ∈ TarskiG) |
16 | | outpasch.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 ∈ 𝑃) |
17 | 16 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑅 ∈ 𝑃) |
18 | | outpasch.q |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ 𝑃) |
19 | 18 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ 𝑃) |
20 | | outpasch.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
21 | 20 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ 𝑃) |
22 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐶)) |
23 | | outpasch.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝑅)) |
24 | 8, 9, 10, 11, 1, 20, 16, 23 | tgbtwncom 25183 |
. . . . . . . 8
⊢ (𝜑 → 𝐶 ∈ (𝑅𝐼𝐴)) |
25 | 24 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
26 | 8, 9, 10, 15, 17, 19, 21, 2, 22, 25 | tgbtwnexch 25193 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐴)) |
27 | 14, 26 | jca 553 |
. . . . 5
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴))) |
28 | 2, 7, 27 | rspcedvd 3289 |
. . . 4
⊢ ((𝜑 ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
29 | 28 | adantlr 747 |
. . 3
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
30 | 12 | ad2antrr 758 |
. . . 4
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ 𝑃) |
31 | | eleq1 2676 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐵 ∈ (𝐴𝐼𝐵))) |
32 | | eqidd 2611 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → 𝑄 = 𝑄) |
33 | | oveq2 6557 |
. . . . . . 7
⊢ (𝑥 = 𝐵 → (𝑅𝐼𝑥) = (𝑅𝐼𝐵)) |
34 | 32, 33 | eleq12d 2682 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐵))) |
35 | 31, 34 | anbi12d 743 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
36 | 35 | adantl 481 |
. . . 4
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
37 | 8, 9, 10, 11, 1, 12 | tgbtwntriv2 25182 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐵)) |
38 | 37 | ad2antrr 758 |
. . . . 5
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐵)) |
39 | 11 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐺 ∈ TarskiG) |
40 | 39 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐺 ∈ TarskiG) |
41 | 20 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐶 ∈ 𝑃) |
42 | 16 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ 𝑃) |
43 | 18 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ 𝑃) |
44 | 30 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐵 ∈ 𝑃) |
45 | | simpr 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝑄𝐼𝐶)) |
46 | 8, 9, 10, 40, 43, 42, 41, 45 | tgbtwncom 25183 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝑄)) |
47 | | outpasch.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑄 ∈ (𝐵𝐼𝐶)) |
48 | 8, 9, 10, 11, 12, 18, 20, 47 | tgbtwncom 25183 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈ (𝐶𝐼𝐵)) |
49 | 48 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
50 | 8, 9, 10, 40, 41, 42, 43, 44, 46, 49 | tgbtwnexch3 25189 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
51 | 39 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐺 ∈ TarskiG) |
52 | 30 | adantr 480 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐵 ∈ 𝑃) |
53 | 18 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ 𝑃) |
54 | 16 | ad3antrrr 762 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑅 ∈ 𝑃) |
55 | 20 | ad3antrrr 762 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ 𝑃) |
56 | | simpr 476 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 = 𝐶) |
57 | 8, 9, 10, 11, 16, 20 | tgbtwntriv2 25182 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐶 ∈ (𝑅𝐼𝐶)) |
58 | 57 | ad4antr 764 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝐶 ∈ (𝑅𝐼𝐶)) |
59 | 56, 58 | eqeltrd 2688 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 ∈ (𝑅𝐼𝐶)) |
60 | | simpllr 795 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → ¬ 𝑄 ∈ (𝑅𝐼𝐶)) |
61 | 59, 60 | pm2.65da 598 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → ¬ 𝑄 = 𝐶) |
62 | 61 | neqned 2789 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ≠ 𝐶) |
63 | 47 | ad3antrrr 762 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝐶)) |
64 | | simpr 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ (𝑄𝐼𝑅)) |
65 | 8, 9, 10, 51, 52, 53, 55, 54, 62, 63, 64 | tgbtwnouttr 25192 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝑅)) |
66 | 8, 9, 10, 51, 52, 53, 54, 65 | tgbtwncom 25183 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
67 | | outpasch.l |
. . . . . . . . . . 11
⊢ 𝐿 = (LineG‘𝐺) |
68 | 8, 67, 10, 11, 18, 20, 16 | tgcolg 25249 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
69 | 68 | biimpa 500 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
70 | | 3orcoma 1039 |
. . . . . . . . . 10
⊢ ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
71 | | 3orass 1034 |
. . . . . . . . . 10
⊢ ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
72 | 70, 71 | bitr3i 265 |
. . . . . . . . 9
⊢ ((𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
73 | 69, 72 | sylib 207 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
74 | 73 | ord 391 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (¬ 𝑄 ∈ (𝑅𝐼𝐶) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))) |
75 | 74 | imp 444 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))) |
76 | 50, 66, 75 | mpjaodan 823 |
. . . . 5
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
77 | 38, 76 | jca 553 |
. . . 4
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))) |
78 | 30, 36, 77 | rspcedvd 3289 |
. . 3
⊢ (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
79 | 29, 78 | pm2.61dan 828 |
. 2
⊢ ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
80 | 12 | ad2antrr 758 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ 𝑃) |
81 | 35 | adantl 481 |
. . . 4
⊢ ((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))) |
82 | 37 | ad2antrr 758 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐴𝐼𝐵)) |
83 | 11 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG) |
84 | 16 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ 𝑃) |
85 | 18 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ 𝑃) |
86 | 20 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ 𝑃) |
87 | | simplr 788 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
88 | | simpr 476 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐿𝑄)) |
89 | 11 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐺 ∈ TarskiG) |
90 | 16 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ 𝑃) |
91 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ 𝑃) |
92 | 20 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ∈ 𝑃) |
93 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
94 | 8, 10, 67, 89, 90, 91, 92, 93 | ncolne1 25320 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ≠ 𝑄) |
95 | 8, 10, 67, 89, 90, 91, 94 | tglinerflx2 25329 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
96 | 95 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
97 | 8, 67, 10, 89, 91, 92, 90, 93 | ncolcom 25256 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝐶𝐿𝑄) ∨ 𝐶 = 𝑄)) |
98 | 8, 67, 10, 89, 92, 91, 90, 97 | ncolrot1 25257 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅)) |
99 | 8, 10, 67, 89, 92, 91, 90, 98 | ncolne1 25320 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ≠ 𝑄) |
100 | 99 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ≠ 𝑄) |
101 | 48 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
102 | 8, 10, 67, 83, 86, 85, 80, 100, 101 | btwnlng3 25316 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐶𝐿𝑄)) |
103 | 8, 10, 67, 83, 86, 85, 100 | tglinerflx2 25329 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐿𝑄)) |
104 | 8, 10, 67, 83, 84, 85, 86, 85, 87, 88, 96, 102, 103 | tglineinteq 25340 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 = 𝑄) |
105 | 8, 9, 10, 11, 16, 12 | tgbtwntriv2 25182 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝑅𝐼𝐵)) |
106 | 105 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐼𝐵)) |
107 | 104, 106 | eqeltrrd 2689 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐼𝐵)) |
108 | 82, 107 | jca 553 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))) |
109 | 80, 81, 108 | rspcedvd 3289 |
. . 3
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
110 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑡 = 𝑥 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑥 ∈ (𝑎𝐼𝑏))) |
111 | 110 | cbvrexv 3148 |
. . . . . . . . 9
⊢
(∃𝑡 ∈
(𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏)) |
112 | 111 | anbi2i 726 |
. . . . . . . 8
⊢ (((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))) |
113 | 112 | opabbii 4649 |
. . . . . . 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))} |
114 | 89 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG) |
115 | 90 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ 𝑃) |
116 | 91 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ 𝑃) |
117 | 94 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ≠ 𝑄) |
118 | 8, 10, 67, 114, 115, 116, 117 | tgelrnln 25325 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝑅𝐿𝑄) ∈ ran 𝐿) |
119 | | eqid 2610 |
. . . . . . 7
⊢
(hlG‘𝐺) =
(hlG‘𝐺) |
120 | 20 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ 𝑃) |
121 | 1 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴 ∈ 𝑃) |
122 | 12 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ 𝑃) |
123 | 122 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ 𝑃) |
124 | 95 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄)) |
125 | 8, 67, 10, 89, 91, 92, 90, 93 | ncolrot2 25258 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄)) |
126 | | pm2.45 411 |
. . . . . . . . . 10
⊢ (¬
(𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
127 | 125, 126 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
128 | 127 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄)) |
129 | | simpr 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐵 ∈ (𝑅𝐿𝑄)) |
130 | 48 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵)) |
131 | 8, 9, 10, 113, 120, 123, 124, 128, 129, 130 | islnoppd 25432 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵) |
132 | 8, 10, 67, 89, 90, 91, 94 | tglinerflx1 25328 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ (𝑅𝐿𝑄)) |
133 | 132 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ (𝑅𝐿𝑄)) |
134 | 23 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ (𝐴𝐼𝑅)) |
135 | 24 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
136 | 8, 10, 67, 89, 92, 90, 91, 125 | ncolne1 25320 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶 ≠ 𝑅) |
137 | 136 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ≠ 𝑅) |
138 | 8, 9, 10, 114, 115, 120, 121, 135, 137 | tgbtwnne 25185 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ≠ 𝐴) |
139 | 138 | necomd 2837 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴 ≠ 𝑅) |
140 | 8, 10, 119, 121, 115, 120, 114, 121, 134, 139, 137 | btwnhl2 25308 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶((hlG‘𝐺)‘𝑅)𝐴) |
141 | 8, 9, 10, 113, 67, 118, 114, 119, 120, 121, 123, 131, 133, 140 | opphl 25446 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴{〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵) |
142 | 8, 9, 10, 113, 121, 123 | islnopp 25431 |
. . . . . 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)))) |
143 | 141, 142 | mpbid 221 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵))) |
144 | 143 | simprd 478 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵)) |
145 | 114 | ad2antrr 758 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG) |
146 | 118 | ad2antrr 758 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑅𝐿𝑄) ∈ ran 𝐿) |
147 | | simplr 788 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝑅𝐿𝑄)) |
148 | 8, 67, 10, 145, 146, 147 | tglnpt 25244 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ 𝑃) |
149 | | simpr 476 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐴𝐼𝐵)) |
150 | 145 | ad2antrr 758 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG) |
151 | 90 | ad3antrrr 762 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑅 ∈ 𝑃) |
152 | 151 | ad2antrr 758 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅 ∈ 𝑃) |
153 | 91 | ad5antr 766 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ 𝑃) |
154 | 120 | ad2antrr 758 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ 𝑃) |
155 | 154 | ad2antrr 758 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶 ∈ 𝑃) |
156 | 93 | ad5antr 766 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) |
157 | | simplr 788 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ 𝑃) |
158 | 117 | ad4antr 764 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅 ≠ 𝑄) |
159 | 148 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ 𝑃) |
160 | 94 | necomd 2837 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ≠ 𝑅) |
161 | 160 | ad5antr 766 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ≠ 𝑅) |
162 | 147 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑅𝐿𝑄)) |
163 | 8, 10, 67, 150, 153, 152, 159, 161, 162 | lncom 25317 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑄𝐿𝑅)) |
164 | | simprl 790 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑥𝐼𝑅)) |
165 | 8, 10, 67, 150, 159, 153, 152, 157, 163, 164 | coltr3 25343 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝑅)) |
166 | 8, 10, 67, 150, 152, 153, 157, 158, 165 | lncom 25317 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐿𝑄)) |
167 | 95 | ad5antr 766 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐿𝑄)) |
168 | 99 | ad5antr 766 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶 ≠ 𝑄) |
169 | 123 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃) |
170 | 169 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ 𝑃) |
171 | 99 | necomd 2837 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ≠ 𝐶) |
172 | 47 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐵𝐼𝐶)) |
173 | 8, 10, 67, 89, 91, 92, 122, 171, 172 | btwnlng2 25315 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ (𝑄𝐿𝐶)) |
174 | 173 | ad5antr 766 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ (𝑄𝐿𝐶)) |
175 | | simprr 792 |
. . . . . . . . . . . . . 14
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐼𝐵)) |
176 | 8, 9, 10, 150, 155, 157, 170, 175 | tgbtwncom 25183 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐵𝐼𝐶)) |
177 | 8, 10, 67, 150, 170, 153, 155, 157, 174, 176 | coltr3 25343 |
. . . . . . . . . . . 12
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝐶)) |
178 | 8, 10, 67, 150, 155, 153, 157, 168, 177 | lncom 25317 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐿𝑄)) |
179 | 8, 10, 67, 89, 92, 91, 99 | tglinerflx2 25329 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐶𝐿𝑄)) |
180 | 179 | ad5antr 766 |
. . . . . . . . . . 11
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝐶𝐿𝑄)) |
181 | 8, 10, 67, 150, 152, 153, 155, 153, 156, 166, 167, 178, 180 | tglineinteq 25340 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 = 𝑄) |
182 | 8, 9, 10, 150, 159, 157, 152, 164 | tgbtwncom 25183 |
. . . . . . . . . 10
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐼𝑥)) |
183 | 181, 182 | eqeltrrd 2689 |
. . . . . . . . 9
⊢
(((((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡 ∈ 𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐼𝑥)) |
184 | 121 | ad2antrr 758 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃) |
185 | 8, 9, 10, 145, 184, 148, 169, 149 | tgbtwncom 25183 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐵𝐼𝐴)) |
186 | 24 | ad4antr 764 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝑅𝐼𝐴)) |
187 | 8, 9, 10, 145, 169, 151, 184, 148, 154, 185, 186 | axtgpasch 25166 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → ∃𝑡 ∈ 𝑃 (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) |
188 | 183, 187 | r19.29a 3060 |
. . . . . . . 8
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑄 ∈ (𝑅𝐼𝑥)) |
189 | 148, 149,
188 | jca32 556 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
(𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))) |
190 | 189 | anasss 677 |
. . . . . 6
⊢ ((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ (𝑥 ∈ (𝑅𝐿𝑄) ∧ 𝑥 ∈ (𝐴𝐼𝐵))) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))) |
191 | 190 | ex 449 |
. . . . 5
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((𝑥 ∈ (𝑅𝐿𝑄) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥 ∈ 𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))))) |
192 | 191 | reximdv2 2997 |
. . . 4
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))) |
193 | 144, 192 | mpd 15 |
. . 3
⊢ (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
194 | 109, 193 | pm2.61dan 828 |
. 2
⊢ ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |
195 | 79, 194 | pm2.61dan 828 |
1
⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))) |