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Mirrors > Home > MPE Home > Th. List > umgr3cyclex | Structured version Visualization version GIF version |
Description: If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.) |
Ref | Expression |
---|---|
uhgr3cyclex.v | β’ π = (VtxβπΊ) |
uhgr3cyclex.e | β’ πΈ = (EdgβπΊ) |
Ref | Expression |
---|---|
umgr3cyclex | β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3 β§ (πβ0) = π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgruhgr 27519 | . . 3 β’ (πΊ β UMGraph β πΊ β UHGraph) | |
2 | 1 | 3ad2ant1 1133 | . 2 β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β πΊ β UHGraph) |
3 | simp2 1137 | . 2 β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β (π΄ β π β§ π΅ β π β§ πΆ β π)) | |
4 | uhgr3cyclex.e | . . . . . 6 β’ πΈ = (EdgβπΊ) | |
5 | 4 | umgredgne 27560 | . . . . 5 β’ ((πΊ β UMGraph β§ {π΄, π΅} β πΈ) β π΄ β π΅) |
6 | 5 | 3ad2antr1 1188 | . . . 4 β’ ((πΊ β UMGraph β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β π΄ β π΅) |
7 | prcom 4672 | . . . . . . . 8 β’ {πΆ, π΄} = {π΄, πΆ} | |
8 | 7 | eleq1i 2827 | . . . . . . 7 β’ ({πΆ, π΄} β πΈ β {π΄, πΆ} β πΈ) |
9 | 8 | biimpi 215 | . . . . . 6 β’ ({πΆ, π΄} β πΈ β {π΄, πΆ} β πΈ) |
10 | 9 | 3ad2ant3 1135 | . . . . 5 β’ (({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ) β {π΄, πΆ} β πΈ) |
11 | 4 | umgredgne 27560 | . . . . 5 β’ ((πΊ β UMGraph β§ {π΄, πΆ} β πΈ) β π΄ β πΆ) |
12 | 10, 11 | sylan2 594 | . . . 4 β’ ((πΊ β UMGraph β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β π΄ β πΆ) |
13 | simp2 1137 | . . . . 5 β’ (({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ) β {π΅, πΆ} β πΈ) | |
14 | 4 | umgredgne 27560 | . . . . 5 β’ ((πΊ β UMGraph β§ {π΅, πΆ} β πΈ) β π΅ β πΆ) |
15 | 13, 14 | sylan2 594 | . . . 4 β’ ((πΊ β UMGraph β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β π΅ β πΆ) |
16 | 6, 12, 15 | 3jca 1128 | . . 3 β’ ((πΊ β UMGraph β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) |
17 | 16 | 3adant2 1131 | . 2 β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) |
18 | simp3 1138 | . 2 β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) | |
19 | uhgr3cyclex.v | . . 3 β’ π = (VtxβπΊ) | |
20 | 19, 4 | uhgr3cyclex 28591 | . 2 β’ ((πΊ β UHGraph β§ ((π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π΄ β π΅ β§ π΄ β πΆ β§ π΅ β πΆ)) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3 β§ (πβ0) = π΄)) |
21 | 2, 3, 17, 18, 20 | syl121anc 1375 | 1 β’ ((πΊ β UMGraph β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ ({π΄, π΅} β πΈ β§ {π΅, πΆ} β πΈ β§ {πΆ, π΄} β πΈ)) β βπβπ(π(CyclesβπΊ)π β§ (β―βπ) = 3 β§ (πβ0) = π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1087 = wceq 1539 βwex 1779 β wcel 2104 β wne 2941 {cpr 4567 class class class wbr 5081 βcfv 6458 0cc0 10917 3c3 12075 β―chash 14090 Vtxcvtx 27411 Edgcedg 27462 UHGraphcuhgr 27471 UMGraphcumgr 27496 Cyclesccycls 28198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993 ax-pre-mulgt0 10994 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-tp 4570 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-riota 7264 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-1st 7863 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-1o 8328 df-oadd 8332 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-dju 9703 df-card 9741 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-sub 11253 df-neg 11254 df-nn 12020 df-2 12082 df-3 12083 df-4 12084 df-n0 12280 df-z 12366 df-uz 12629 df-fz 13286 df-fzo 13429 df-hash 14091 df-word 14263 df-concat 14319 df-s1 14346 df-s2 14606 df-s3 14607 df-s4 14608 df-edg 27463 df-uhgr 27473 df-upgr 27497 df-umgr 27498 df-wlks 28011 df-trls 28105 df-pths 28129 df-cycls 28200 |
This theorem is referenced by: umgr3v3e3cycl 28593 3cyclfrgr 28697 cusgr3cyclex 33143 |
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