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Theorem konigsberglem1 29199
Description: Lemma 1 for konigsberg 29204: Vertex 0 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.v 𝑉 = (0...3)
konigsberg.e 𝐸 = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
konigsberg.g 𝐺 = βŸ¨π‘‰, 𝐸⟩
Assertion
Ref Expression
konigsberglem1 ((VtxDegβ€˜πΊ)β€˜0) = 3

Proof of Theorem konigsberglem1
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ovex 7391 . . . 4 (0...3) ∈ V
2 s6cli 14774 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word V
32elexi 3465 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ V
41, 3opvtxfvi 27963 . . 3 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = (0...3)
54eqcomi 2746 . 2 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)
6 3nn0 12432 . . 3 3 ∈ β„•0
7 0elfz 13539 . . 3 (3 ∈ β„•0 β†’ 0 ∈ (0...3))
86, 7ax-mp 5 . 2 0 ∈ (0...3)
91, 3opiedgfvi 27964 . . 3 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©
109eqcomi 2746 . 2 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)
11 s1cli 14494 . . 3 βŸ¨β€œ{2, 3}β€βŸ© ∈ Word V
12 df-s7 14743 . . 3 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
13 eqid 2737 . . . 4 (0...3) = (0...3)
14 eqid 2737 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
15 eqid 2737 . . . 4 ⟨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©βŸ© = ⟨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©βŸ©
1613, 14, 15konigsbergssiedgw 29197 . . 3 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
172, 11, 12, 16mp3an 1462 . 2 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
18 s5cli 14773 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word V
1918elexi 3465 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ V
201, 19opvtxfvi 27963 . . . 4 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = (0...3)
2120eqcomi 2746 . . 3 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)
221, 19opiedgfvi 27964 . . . 4 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©
2322eqcomi 2746 . . 3 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)
24 s2cli 14770 . . . 4 βŸ¨β€œ{2, 3} {2, 3}β€βŸ© ∈ Word V
25 s5s2 14825 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3} {2, 3}β€βŸ©)
2613, 14, 15konigsbergssiedgw 29197 . . . 4 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
2718, 24, 25, 26mp3an 1462 . . 3 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
28 s4cli 14772 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word V
2928elexi 3465 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ V
301, 29opvtxfvi 27963 . . . . 5 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = (0...3)
3130eqcomi 2746 . . . 4 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)
321, 29opiedgfvi 27964 . . . . 5 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©
3332eqcomi 2746 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)
34 s3cli 14771 . . . . 5 βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
35 s4s3 14821 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ©)
3613, 14, 15konigsbergssiedgw 29197 . . . . 5 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
3728, 34, 35, 36mp3an 1462 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
38 s3cli 14771 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word V
3938elexi 3465 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ V
401, 39opvtxfvi 27963 . . . . . 6 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = (0...3)
4140eqcomi 2746 . . . . 5 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)
421, 39opiedgfvi 27964 . . . . . 6 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©
4342eqcomi 2746 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)
44 s4cli 14772 . . . . . 6 βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
45 s3s4 14823 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
4613, 14, 15konigsbergssiedgw 29197 . . . . . 6 ((βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
4738, 44, 45, 46mp3an 1462 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
48 s2cli 14770 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word V
4948elexi 3465 . . . . . . . . 9 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ V
501, 49opvtxfvi 27963 . . . . . . . 8 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = (0...3)
5150eqcomi 2746 . . . . . . 7 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
521, 49opiedgfvi 27964 . . . . . . . 8 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1} {0, 2}β€βŸ©
5352eqcomi 2746 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)
54 s5cli 14773 . . . . . . . 8 βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
55 s2s5 14824 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
5613, 14, 15konigsbergssiedgw 29197 . . . . . . . 8 ((βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
5748, 54, 55, 56mp3an 1462 . . . . . . 7 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
58 s1cli 14494 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word V
5958elexi 3465 . . . . . . . . . . 11 βŸ¨β€œ{0, 1}β€βŸ© ∈ V
601, 59opvtxfvi 27963 . . . . . . . . . 10 (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (0...3)
6160eqcomi 2746 . . . . . . . . 9 (0...3) = (Vtxβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
621, 59opiedgfvi 27964 . . . . . . . . . 10 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = βŸ¨β€œ{0, 1}β€βŸ©
6362eqcomi 2746 . . . . . . . . 9 βŸ¨β€œ{0, 1}β€βŸ© = (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)
64 s6cli 14774 . . . . . . . . . 10 βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V
65 s1s6 14817 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)
6613, 14, 15konigsbergssiedgw 29197 . . . . . . . . . 10 ((βŸ¨β€œ{0, 1}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© ∈ Word V ∧ βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©)) β†’ βŸ¨β€œ{0, 1}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2})
6758, 64, 65, 66mp3an 1462 . . . . . . . . 9 βŸ¨β€œ{0, 1}β€βŸ© ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
68 0ex 5265 . . . . . . . . . . . . 13 βˆ… ∈ V
691, 68opvtxfvi 27963 . . . . . . . . . . . 12 (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©) = (0...3)
7069eqcomi 2746 . . . . . . . . . . 11 (0...3) = (Vtxβ€˜βŸ¨(0...3), βˆ…βŸ©)
711, 68opiedgfvi 27964 . . . . . . . . . . . 12 (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©) = βˆ…
7271eqcomi 2746 . . . . . . . . . . 11 βˆ… = (iEdgβ€˜βŸ¨(0...3), βˆ…βŸ©)
73 wrd0 14428 . . . . . . . . . . 11 βˆ… ∈ Word {π‘₯ ∈ (𝒫 (0...3) βˆ– {βˆ…}) ∣ (β™―β€˜π‘₯) ≀ 2}
74 eqid 2737 . . . . . . . . . . . 12 βˆ… = βˆ…
7570, 72vtxdg0e 28425 . . . . . . . . . . . 12 ((0 ∈ (0...3) ∧ βˆ… = βˆ…) β†’ ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜0) = 0)
768, 74, 75mp2an 691 . . . . . . . . . . 11 ((VtxDegβ€˜βŸ¨(0...3), βˆ…βŸ©)β€˜0) = 0
77 1nn0 12430 . . . . . . . . . . . 12 1 ∈ β„•0
78 1le3 12366 . . . . . . . . . . . 12 1 ≀ 3
79 elfz2nn0 13533 . . . . . . . . . . . 12 (1 ∈ (0...3) ↔ (1 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 1 ≀ 3))
8077, 6, 78, 79mpbir3an 1342 . . . . . . . . . . 11 1 ∈ (0...3)
81 ax-1ne0 11121 . . . . . . . . . . 11 1 β‰  0
82 s0s1 14812 . . . . . . . . . . . 12 βŸ¨β€œ{0, 1}β€βŸ© = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
8362, 82eqtri 2765 . . . . . . . . . . 11 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©) = (βˆ… ++ βŸ¨β€œ{0, 1}β€βŸ©)
8470, 8, 72, 73, 76, 60, 80, 81, 83vdegp1bi 28488 . . . . . . . . . 10 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)β€˜0) = (0 + 1)
85 0p1e1 12276 . . . . . . . . . 10 (0 + 1) = 1
8684, 85eqtri 2765 . . . . . . . . 9 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1}β€βŸ©βŸ©)β€˜0) = 1
87 2nn0 12431 . . . . . . . . . 10 2 ∈ β„•0
88 2re 12228 . . . . . . . . . . 11 2 ∈ ℝ
89 3re 12234 . . . . . . . . . . 11 3 ∈ ℝ
90 2lt3 12326 . . . . . . . . . . 11 2 < 3
9188, 89, 90ltleii 11279 . . . . . . . . . 10 2 ≀ 3
92 elfz2nn0 13533 . . . . . . . . . 10 (2 ∈ (0...3) ↔ (2 ∈ β„•0 ∧ 3 ∈ β„•0 ∧ 2 ≀ 3))
9387, 6, 91, 92mpbir3an 1342 . . . . . . . . 9 2 ∈ (0...3)
94 2ne0 12258 . . . . . . . . 9 2 β‰  0
95 df-s2 14738 . . . . . . . . . 10 βŸ¨β€œ{0, 1} {0, 2}β€βŸ© = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2}β€βŸ©)
9652, 95eqtri 2765 . . . . . . . . 9 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1}β€βŸ© ++ βŸ¨β€œ{0, 2}β€βŸ©)
9761, 8, 63, 67, 86, 50, 93, 94, 96vdegp1bi 28488 . . . . . . . 8 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)β€˜0) = (1 + 1)
98 1p1e2 12279 . . . . . . . 8 (1 + 1) = 2
9997, 98eqtri 2765 . . . . . . 7 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2}β€βŸ©βŸ©)β€˜0) = 2
100 nn0fz0 13540 . . . . . . . 8 (3 ∈ β„•0 ↔ 3 ∈ (0...3))
1016, 100mpbi 229 . . . . . . 7 3 ∈ (0...3)
102 3ne0 12260 . . . . . . 7 3 β‰  0
103 df-s3 14739 . . . . . . . 8 βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3}β€βŸ©)
10442, 103eqtri 2765 . . . . . . 7 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2}β€βŸ© ++ βŸ¨β€œ{0, 3}β€βŸ©)
10551, 8, 53, 57, 99, 40, 101, 102, 104vdegp1bi 28488 . . . . . 6 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)β€˜0) = (2 + 1)
106 2p1e3 12296 . . . . . 6 (2 + 1) = 3
107105, 106eqtri 2765 . . . . 5 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ©βŸ©)β€˜0) = 3
108 df-s4 14740 . . . . . 6 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
10932, 108eqtri 2765 . . . . 5 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
11041, 8, 43, 47, 107, 30, 80, 81, 93, 94, 109vdegp1ai 28487 . . . 4 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ©βŸ©)β€˜0) = 3
111 df-s5 14741 . . . . 5 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
11222, 111eqtri 2765 . . . 4 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2}β€βŸ© ++ βŸ¨β€œ{1, 2}β€βŸ©)
11331, 8, 33, 37, 110, 20, 80, 81, 93, 94, 112vdegp1ai 28487 . . 3 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ©βŸ©)β€˜0) = 3
114 df-s6 14742 . . . 4 βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
1159, 114eqtri 2765 . . 3 (iEdgβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
11621, 8, 23, 27, 113, 4, 93, 94, 101, 102, 115vdegp1ai 28487 . 2 ((VtxDegβ€˜βŸ¨(0...3), βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ©βŸ©)β€˜0) = 3
117 konigsberg.v . . 3 𝑉 = (0...3)
118 konigsberg.e . . 3 𝐸 = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
119 konigsberg.g . . 3 𝐺 = βŸ¨π‘‰, 𝐸⟩
120117, 118, 119konigsbergvtx 29193 . 2 (Vtxβ€˜πΊ) = (0...3)
121117, 118, 119konigsbergiedg 29194 . . 3 (iEdgβ€˜πΊ) = βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}β€βŸ©
122121, 12eqtri 2765 . 2 (iEdgβ€˜πΊ) = (βŸ¨β€œ{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3}β€βŸ© ++ βŸ¨β€œ{2, 3}β€βŸ©)
1235, 8, 10, 17, 116, 120, 93, 94, 101, 102, 122vdegp1ai 28487 1 ((VtxDegβ€˜πΊ)β€˜0) = 3
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  {crab 3408  Vcvv 3446   βˆ– cdif 3908  βˆ…c0 4283  π’« cpw 4561  {csn 4587  {cpr 4589  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  0cc0 11052  1c1 11053   + caddc 11055   ≀ cle 11191  2c2 12209  3c3 12210  β„•0cn0 12414  ...cfz 13425  β™―chash 14231  Word cword 14403   ++ cconcat 14459  βŸ¨β€œcs1 14484  βŸ¨β€œcs2 14731  βŸ¨β€œcs3 14732  βŸ¨β€œcs4 14733  βŸ¨β€œcs5 14734  βŸ¨β€œcs6 14735  βŸ¨β€œcs7 14736  Vtxcvtx 27950  iEdgciedg 27951  VtxDegcvtxdg 28416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9838  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-xnn0 12487  df-z 12501  df-uz 12765  df-xadd 13035  df-fz 13426  df-fzo 13569  df-hash 14232  df-word 14404  df-concat 14460  df-s1 14485  df-s2 14738  df-s3 14739  df-s4 14740  df-s5 14741  df-s6 14742  df-s7 14743  df-vtx 27952  df-iedg 27953  df-vtxdg 28417
This theorem is referenced by:  konigsberglem4  29202
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