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Mirrors > Home > MPE Home > Th. List > konigsberg | Structured version Visualization version GIF version |
Description: The Königsberg Bridge problem. If 𝐺 is the Königsberg graph, i.e. a graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpath 27101 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.) |
Ref | Expression |
---|---|
konigsberg.v | ⊢ 𝑉 = (0...3) |
konigsberg.e | ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 |
konigsberg.g | ⊢ 𝐺 = 〈𝑉, 𝐸〉 |
Ref | Expression |
---|---|
konigsberg | ⊢ (EulerPaths‘𝐺) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | konigsberg.v | . . . 4 ⊢ 𝑉 = (0...3) | |
2 | konigsberg.e | . . . 4 ⊢ 𝐸 = 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 | |
3 | konigsberg.g | . . . 4 ⊢ 𝐺 = 〈𝑉, 𝐸〉 | |
4 | 1, 2, 3 | konigsberglem5 27118 | . . 3 ⊢ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
5 | elpri 4197 | . . . 4 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2)) | |
6 | 2pos 11112 | . . . . . . 7 ⊢ 0 < 2 | |
7 | 0re 10040 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 2re 11090 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 7, 8 | ltnsymi 10156 | . . . . . . 7 ⊢ (0 < 2 → ¬ 2 < 0) |
10 | 6, 9 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 < 0 |
11 | breq2 4657 | . . . . . 6 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0)) | |
12 | 10, 11 | mtbiri 317 | . . . . 5 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
13 | 8 | ltnri 10146 | . . . . . 6 ⊢ ¬ 2 < 2 |
14 | breq2 4657 | . . . . . 6 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2)) | |
15 | 13, 14 | mtbiri 317 | . . . . 5 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
16 | 12, 15 | jaoi 394 | . . . 4 ⊢ (((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2) → ¬ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
17 | 5, 16 | syl 17 | . . 3 ⊢ ((#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ¬ 2 < (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
18 | 4, 17 | mt2 191 | . 2 ⊢ ¬ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} |
19 | 1, 2, 3 | konigsbergumgr 27112 | . . . . 5 ⊢ 𝐺 ∈ UMGraph |
20 | umgrupgr 25998 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph ) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ 𝐺 ∈ UPGraph |
22 | 3 | fveq2i 6194 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, 𝐸〉) |
23 | ovex 6678 | . . . . . . . 8 ⊢ (0...3) ∈ V | |
24 | 1, 23 | eqeltri 2697 | . . . . . . 7 ⊢ 𝑉 ∈ V |
25 | s7cli 13630 | . . . . . . . 8 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word V | |
26 | 2, 25 | eqeltri 2697 | . . . . . . 7 ⊢ 𝐸 ∈ Word V |
27 | opvtxfv 25884 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
28 | 24, 26, 27 | mp2an 708 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
29 | 22, 28 | eqtr2i 2645 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) |
30 | 29 | eulerpath 27101 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
31 | 21, 30 | mpan 706 | . . 3 ⊢ ((EulerPaths‘𝐺) ≠ ∅ → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
32 | 31 | necon1bi 2822 | . 2 ⊢ (¬ (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → (EulerPaths‘𝐺) = ∅) |
33 | 18, 32 | ax-mp 5 | 1 ⊢ (EulerPaths‘𝐺) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 383 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∅c0 3915 {cpr 4179 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 < clt 10074 2c2 11070 3c3 11071 ...cfz 12326 #chash 13117 Word cword 13291 〈“cs7 13591 ∥ cdvds 14983 Vtxcvtx 25874 UPGraph cupgr 25975 UMGraph cumgr 25976 VtxDegcvtxdg 26361 EulerPathsceupth 27057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-s4 13595 df-s5 13596 df-s6 13597 df-s7 13598 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-umgr 25978 df-uspgr 26045 df-vtxdg 26362 df-wlks 26495 df-trls 26589 df-eupth 27058 |
This theorem is referenced by: (None) |
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