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Theorem konigsberglem3 41424
 Description: Lemma 3 for konigsberg-av 41427: Vertex 3 has degree three. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 4-Mar-2021.)
Hypotheses
Ref Expression
konigsberg-av.v 𝑉 = (0...3)
konigsberg-av.e 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
konigsberg-av.g 𝐺 = ⟨𝑉, 𝐸
Assertion
Ref Expression
konigsberglem3 ((VtxDeg‘𝐺)‘3) = 3

Proof of Theorem konigsberglem3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . . 5 (0...3) ∈ V
2 s6cli 13479 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V
32elexi 3186 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ V
41, 3opvtxfvi 25686 . . . 4 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (0...3)
54eqcomi 2619 . . 3 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
6 3nn0 11187 . . . 4 3 ∈ ℕ0
7 nn0fz0 12306 . . . 4 (3 ∈ ℕ0 ↔ 3 ∈ (0...3))
86, 7mpbi 219 . . 3 3 ∈ (0...3)
91, 3opiedgfvi 25687 . . . 4 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩
109eqcomi 2619 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)
11 s1cli 13237 . . . 4 ⟨“{2, 3}”⟩ ∈ Word V
12 df-s7 13449 . . . 4 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
13 eqid 2610 . . . . 5 (0...3) = (0...3)
14 eqid 2610 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
15 eqid 2610 . . . . 5 ⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩⟩ = ⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩⟩
1613, 14, 15konigsbergssiedgw 41419 . . . 4 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word V ∧ ⟨“{2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
172, 11, 12, 16mp3an 1416 . . 3 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
18 s5cli 13478 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V
1918elexi 3186 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ V
201, 19opvtxfvi 25686 . . . . . 6 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (0...3)
2120eqcomi 2619 . . . . 5 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
221, 19opiedgfvi 25687 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩
2322eqcomi 2619 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)
24 s2cli 13475 . . . . . 6 ⟨“{2, 3} {2, 3}”⟩ ∈ Word V
25 s5s2 13530 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3} {2, 3}”⟩)
2613, 14, 15konigsbergssiedgw 41419 . . . . . 6 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word V ∧ ⟨“{2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
2718, 24, 25, 26mp3an 1416 . . . . 5 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
28 s4cli 13477 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V
2928elexi 3186 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ V
301, 29opvtxfvi 25686 . . . . . . 7 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (0...3)
3130eqcomi 2619 . . . . . 6 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
321, 29opiedgfvi 25687 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩
3332eqcomi 2619 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)
34 s3cli 13476 . . . . . . 7 ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
35 s4s3 13526 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2} {2, 3} {2, 3}”⟩)
3613, 14, 15konigsbergssiedgw 41419 . . . . . . 7 ((⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word V ∧ ⟨“{1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
3728, 34, 35, 36mp3an 1416 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
38 s3cli 13476 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V
3938elexi 3186 . . . . . . . . 9 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ V
401, 39opvtxfvi 25686 . . . . . . . 8 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (0...3)
4140eqcomi 2619 . . . . . . 7 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
421, 39opiedgfvi 25687 . . . . . . . 8 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = ⟨“{0, 1} {0, 2} {0, 3}”⟩
4342eqcomi 2619 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)
44 s4cli 13477 . . . . . . . 8 ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
45 s3s4 13528 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩)
4613, 14, 15konigsbergssiedgw 41419 . . . . . . . 8 ((⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word V ∧ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4738, 44, 45, 46mp3an 1416 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
48 s2cli 13475 . . . . . . . . . . . 12 ⟨“{0, 1} {0, 2}”⟩ ∈ Word V
4948elexi 3186 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ ∈ V
501, 49opvtxfvi 25686 . . . . . . . . . 10 (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (0...3)
5150eqcomi 2619 . . . . . . . . 9 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
521, 49opiedgfvi 25687 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = ⟨“{0, 1} {0, 2}”⟩
5352eqcomi 2619 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)
54 s5cli 13478 . . . . . . . . . 10 ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
55 s2s5 13529 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)
5613, 14, 15konigsbergssiedgw 41419 . . . . . . . . . 10 ((⟨“{0, 1} {0, 2}”⟩ ∈ Word V ∧ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
5748, 54, 55, 56mp3an 1416 . . . . . . . . 9 ⟨“{0, 1} {0, 2}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
58 s1cli 13237 . . . . . . . . . . . . 13 ⟨“{0, 1}”⟩ ∈ Word V
5958elexi 3186 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ ∈ V
601, 59opvtxfvi 25686 . . . . . . . . . . 11 (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (0...3)
6160eqcomi 2619 . . . . . . . . . 10 (0...3) = (Vtx‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
621, 59opiedgfvi 25687 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = ⟨“{0, 1}”⟩
6362eqcomi 2619 . . . . . . . . . 10 ⟨“{0, 1}”⟩ = (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)
64 s6cli 13479 . . . . . . . . . . 11 ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
65 s1s6 13522 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)
6613, 14, 15konigsbergssiedgw 41419 . . . . . . . . . . 11 ((⟨“{0, 1}”⟩ ∈ Word V ∧ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V ∧ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩)) → ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
6758, 64, 65, 66mp3an 1416 . . . . . . . . . 10 ⟨“{0, 1}”⟩ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
68 0ex 4718 . . . . . . . . . . . . 13 ∅ ∈ V
691, 68opvtxfvi 25686 . . . . . . . . . . . 12 (Vtx‘⟨(0...3), ∅⟩) = (0...3)
7069eqcomi 2619 . . . . . . . . . . 11 (0...3) = (Vtx‘⟨(0...3), ∅⟩)
711, 68opiedgfvi 25687 . . . . . . . . . . . 12 (iEdg‘⟨(0...3), ∅⟩) = ∅
7271eqcomi 2619 . . . . . . . . . . 11 ∅ = (iEdg‘⟨(0...3), ∅⟩)
73 wrd0 13185 . . . . . . . . . . 11 ∅ ∈ Word {𝑥 ∈ (𝒫 (0...3) ∖ {∅}) ∣ (#‘𝑥) ≤ 2}
74 eqid 2610 . . . . . . . . . . . 12 ∅ = ∅
7570, 72vtxdg0e 40689 . . . . . . . . . . . 12 ((3 ∈ (0...3) ∧ ∅ = ∅) → ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0)
768, 74, 75mp2an 704 . . . . . . . . . . 11 ((VtxDeg‘⟨(0...3), ∅⟩)‘3) = 0
77 0elfz 12305 . . . . . . . . . . . 12 (3 ∈ ℕ0 → 0 ∈ (0...3))
786, 77ax-mp 5 . . . . . . . . . . 11 0 ∈ (0...3)
79 3ne0 10992 . . . . . . . . . . . 12 3 ≠ 0
8079necomi 2836 . . . . . . . . . . 11 0 ≠ 3
81 1nn0 11185 . . . . . . . . . . . 12 1 ∈ ℕ0
82 1le3 11121 . . . . . . . . . . . 12 1 ≤ 3
83 elfz2nn0 12300 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 1 ≤ 3))
8481, 6, 82, 83mpbir3an 1237 . . . . . . . . . . 11 1 ∈ (0...3)
85 1re 9918 . . . . . . . . . . . 12 1 ∈ ℝ
86 1lt3 11073 . . . . . . . . . . . 12 1 < 3
8785, 86ltneii 10029 . . . . . . . . . . 11 1 ≠ 3
88 s0s1 13517 . . . . . . . . . . . 12 ⟨“{0, 1}”⟩ = (∅ ++ ⟨“{0, 1}”⟩)
8962, 88eqtri 2632 . . . . . . . . . . 11 (iEdg‘⟨(0...3), ⟨“{0, 1}”⟩⟩) = (∅ ++ ⟨“{0, 1}”⟩)
9070, 8, 72, 73, 76, 60, 78, 80, 84, 87, 89vdegp1ai-av 40752 . . . . . . . . . 10 ((VtxDeg‘⟨(0...3), ⟨“{0, 1}”⟩⟩)‘3) = 0
91 2nn0 11186 . . . . . . . . . . 11 2 ∈ ℕ0
92 2re 10967 . . . . . . . . . . . 12 2 ∈ ℝ
93 3re 10971 . . . . . . . . . . . 12 3 ∈ ℝ
94 2lt3 11072 . . . . . . . . . . . 12 2 < 3
9592, 93, 94ltleii 10039 . . . . . . . . . . 11 2 ≤ 3
96 elfz2nn0 12300 . . . . . . . . . . 11 (2 ∈ (0...3) ↔ (2 ∈ ℕ0 ∧ 3 ∈ ℕ0 ∧ 2 ≤ 3))
9791, 6, 95, 96mpbir3an 1237 . . . . . . . . . 10 2 ∈ (0...3)
9892, 94ltneii 10029 . . . . . . . . . 10 2 ≠ 3
99 df-s2 13444 . . . . . . . . . . 11 ⟨“{0, 1} {0, 2}”⟩ = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
10052, 99eqtri 2632 . . . . . . . . . 10 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩) = (⟨“{0, 1}”⟩ ++ ⟨“{0, 2}”⟩)
10161, 8, 63, 67, 90, 50, 78, 80, 97, 98, 100vdegp1ai-av 40752 . . . . . . . . 9 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2}”⟩⟩)‘3) = 0
102 df-s3 13445 . . . . . . . . . 10 ⟨“{0, 1} {0, 2} {0, 3}”⟩ = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10342, 102eqtri 2632 . . . . . . . . 9 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩) = (⟨“{0, 1} {0, 2}”⟩ ++ ⟨“{0, 3}”⟩)
10451, 8, 53, 57, 101, 40, 78, 80, 103vdegp1ci-av 40754 . . . . . . . 8 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = (0 + 1)
105 0p1e1 11009 . . . . . . . 8 (0 + 1) = 1
106104, 105eqtri 2632 . . . . . . 7 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3}”⟩⟩)‘3) = 1
107 df-s4 13446 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10832, 107eqtri 2632 . . . . . . 7 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3}”⟩ ++ ⟨“{1, 2}”⟩)
10941, 8, 43, 47, 106, 30, 84, 87, 97, 98, 108vdegp1ai-av 40752 . . . . . 6 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩⟩)‘3) = 1
110 df-s5 13447 . . . . . . 7 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11122, 110eqtri 2632 . . . . . 6 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2}”⟩ ++ ⟨“{1, 2}”⟩)
11231, 8, 33, 37, 109, 20, 84, 87, 97, 98, 111vdegp1ai-av 40752 . . . . 5 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩⟩)‘3) = 1
113 df-s6 13448 . . . . . 6 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
1149, 113eqtri 2632 . . . . 5 (iEdg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}”⟩ ++ ⟨“{2, 3}”⟩)
11521, 8, 23, 27, 112, 4, 97, 98, 114vdegp1ci-av 40754 . . . 4 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = (1 + 1)
116 1p1e2 11011 . . . 4 (1 + 1) = 2
117115, 116eqtri 2632 . . 3 ((VtxDeg‘⟨(0...3), ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩⟩)‘3) = 2
118 konigsberg-av.v . . . 4 𝑉 = (0...3)
119 konigsberg-av.e . . . 4 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
120 konigsberg-av.g . . . 4 𝐺 = ⟨𝑉, 𝐸
121118, 119, 120konigsbergvtx 41414 . . 3 (Vtx‘𝐺) = (0...3)
122118, 119, 120konigsbergiedg 41415 . . . 4 (iEdg‘𝐺) = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
123122, 12eqtri 2632 . . 3 (iEdg‘𝐺) = (⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}”⟩ ++ ⟨“{2, 3}”⟩)
1245, 8, 10, 17, 117, 121, 97, 98, 123vdegp1ci-av 40754 . 2 ((VtxDeg‘𝐺)‘3) = (2 + 1)
125 2p1e3 11028 . 2 (2 + 1) = 3
126124, 125eqtri 2632 1 ((VtxDeg‘𝐺)‘3) = 3
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  2c2 10947  3c3 10948  ℕ0cn0 11169  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149  ⟨“cs2 13437  ⟨“cs3 13438  ⟨“cs4 13439  ⟨“cs5 13440  ⟨“cs6 13441  ⟨“cs7 13442  Vtxcvtx 25673  iEdgciedg 25674  VtxDegcvtxdg 40681 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-s4 13446  df-s5 13447  df-s6 13448  df-s7 13449  df-vtx 25675  df-iedg 25676  df-vtxdg 40682 This theorem is referenced by:  konigsberglem4  41425
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