Step | Hyp | Ref
| Expression |
1 | | cgracol.p |
. . 3
⊢ 𝑃 = (Base‘𝐺) |
2 | | cgracol.i |
. . 3
⊢ 𝐼 = (Itv‘𝐺) |
3 | | cgrahl.k |
. . 3
⊢ 𝐾 = (hlG‘𝐺) |
4 | | cgracol.d |
. . . 4
⊢ (𝜑 → 𝐷 ∈ 𝑃) |
5 | 4 | ad3antrrr 766 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷 ∈ 𝑃) |
6 | | simplr 792 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦 ∈ 𝑃) |
7 | | cgracol.f |
. . . 4
⊢ (𝜑 → 𝐹 ∈ 𝑃) |
8 | 7 | ad3antrrr 766 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐹 ∈ 𝑃) |
9 | | cgracol.g |
. . . 4
⊢ (𝜑 → 𝐺 ∈ TarskiG) |
10 | 9 | ad3antrrr 766 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐺 ∈ TarskiG) |
11 | | cgracol.e |
. . . 4
⊢ (𝜑 → 𝐸 ∈ 𝑃) |
12 | 11 | ad3antrrr 766 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐸 ∈ 𝑃) |
13 | | simpllr 799 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥 ∈ 𝑃) |
14 | | simpr2 1068 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝐸)𝐷) |
15 | 1, 2, 3, 13, 5, 12, 10, 14 | hlcomd 25499 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝑥) |
16 | 1, 2, 3, 13, 5, 12, 10, 14 | hlne1 25500 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥 ≠ 𝐸) |
17 | | simpr3 1069 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦(𝐾‘𝐸)𝐹) |
18 | 1, 2, 3, 6, 8, 12,
10, 17 | hlne1 25500 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑦 ≠ 𝐸) |
19 | | cgracol.m |
. . . . . . . . 9
⊢ − =
(dist‘𝐺) |
20 | | eqid 2622 |
. . . . . . . . 9
⊢
(cgrG‘𝐺) =
(cgrG‘𝐺) |
21 | 10 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG) |
22 | | cgracol.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ 𝑃) |
23 | 22 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃) |
24 | | cgracol.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ 𝑃) |
25 | 24 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ 𝑃) |
26 | | cgracol.c |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 ∈ 𝑃) |
27 | 26 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃) |
28 | 12 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐸 ∈ 𝑃) |
29 | 13 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝑥 ∈ 𝑃) |
30 | 6 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝑦 ∈ 𝑃) |
31 | | simplr1 1103 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉) |
32 | 1, 19, 2, 20, 21, 25, 23, 27, 29, 28, 30, 31 | cgr3swap23 25419 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 〈“𝐴𝐶𝐵”〉(cgrG‘𝐺)〈“𝑥𝑦𝐸”〉) |
33 | 1, 19, 2, 20, 21, 25, 27, 23, 29, 30, 28, 32 | cgr3rotr 25421 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 〈“𝐵𝐴𝐶”〉(cgrG‘𝐺)〈“𝐸𝑥𝑦”〉) |
34 | | simpr 477 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝐵𝐼𝐶)) |
35 | 1, 19, 2, 20, 21, 23, 25, 27, 28, 29, 30, 33, 34 | tgbtwnxfr 25425 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → 𝑥 ∈ (𝐸𝐼𝑦)) |
36 | 35 | orcd 407 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐴 ∈ (𝐵𝐼𝐶)) → (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥))) |
37 | 9 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG) |
38 | 22 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐵 ∈ 𝑃) |
39 | 26 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ 𝑃) |
40 | 24 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐴 ∈ 𝑃) |
41 | 11 | ad4antr 768 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐸 ∈ 𝑃) |
42 | 6 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝑦 ∈ 𝑃) |
43 | 13 | adantr 481 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝑥 ∈ 𝑃) |
44 | | simplr1 1103 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉) |
45 | 1, 19, 2, 20, 37, 40, 38, 39, 43, 41, 42, 44 | cgr3rotl 25422 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 〈“𝐵𝐶𝐴”〉(cgrG‘𝐺)〈“𝐸𝑦𝑥”〉) |
46 | | simpr 477 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝐶 ∈ (𝐵𝐼𝐴)) |
47 | 1, 19, 2, 20, 37, 38, 39, 40, 41, 42, 43, 45, 46 | tgbtwnxfr 25425 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → 𝑦 ∈ (𝐸𝐼𝑥)) |
48 | 47 | olcd 408 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) ∧ 𝐶 ∈ (𝐵𝐼𝐴)) → (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥))) |
49 | | cgrahl.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶) |
50 | 1, 2, 3, 24, 26, 22, 9 | ishlg 25497 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴(𝐾‘𝐵)𝐶 ↔ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))))) |
51 | 49, 50 | mpbid 222 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))) |
52 | 51 | simp3d 1075 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
53 | 52 | ad3antrrr 766 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))) |
54 | 36, 48, 53 | mpjaodan 827 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥))) |
55 | 16, 18, 54 | 3jca 1242 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥 ≠ 𝐸 ∧ 𝑦 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥)))) |
56 | 1, 2, 3, 13, 6, 12, 10 | ishlg 25497 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → (𝑥(𝐾‘𝐸)𝑦 ↔ (𝑥 ≠ 𝐸 ∧ 𝑦 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥))))) |
57 | 55, 56 | mpbird 247 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝑥(𝐾‘𝐸)𝑦) |
58 | 1, 2, 3, 5, 13, 6,
10, 12, 15, 57 | hltr 25505 |
. . 3
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝑦) |
59 | 1, 2, 3, 5, 6, 8, 10, 12, 58, 17 | hltr 25505 |
. 2
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) → 𝐷(𝐾‘𝐸)𝐹) |
60 | | cgracol.1 |
. . 3
⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉) |
61 | 1, 2, 3, 9, 24, 22, 26, 4, 11, 7 | iscgra 25701 |
. . 3
⊢ (𝜑 → (〈“𝐴𝐵𝐶”〉(cgrA‘𝐺)〈“𝐷𝐸𝐹”〉 ↔ ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹))) |
62 | 60, 61 | mpbid 222 |
. 2
⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 (〈“𝐴𝐵𝐶”〉(cgrG‘𝐺)〈“𝑥𝐸𝑦”〉 ∧ 𝑥(𝐾‘𝐸)𝐷 ∧ 𝑦(𝐾‘𝐸)𝐹)) |
63 | 59, 62 | r19.29vva 3081 |
1
⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹) |