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| Mirrors > Home > MPE Home > Th. List > eulercrct | Structured version Visualization version GIF version | ||
| Description: A pseudograph with an Eulerian circuit β¨πΉ, πβ© (an "Eulerian pseudograph") has only vertices of even degree. (Contributed by AV, 12-Mar-2021.) |
| Ref | Expression |
|---|---|
| eulerpathpr.v | β’ π = (VtxβπΊ) |
| Ref | Expression |
|---|---|
| eulercrct | β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eulerpathpr.v | . . . 4 β’ π = (VtxβπΊ) | |
| 2 | eqid 2725 | . . . 4 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 3 | simpl 481 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΊ β UPGraph) | |
| 4 | upgruhgr 28959 | . . . . . 6 β’ (πΊ β UPGraph β πΊ β UHGraph) | |
| 5 | 2 | uhgrfun 28923 | . . . . . 6 β’ (πΊ β UHGraph β Fun (iEdgβπΊ)) |
| 6 | 4, 5 | syl 17 | . . . . 5 β’ (πΊ β UPGraph β Fun (iEdgβπΊ)) |
| 7 | 6 | adantr 479 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β Fun (iEdgβπΊ)) |
| 8 | simpr 483 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΉ(EulerPathsβπΊ)π) | |
| 9 | 1, 2, 3, 7, 8 | eupth2 30093 | . . 3 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))})) |
| 10 | 9 | 3adant3 1129 | . 2 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))})) |
| 11 | crctprop 29650 | . . . . . . 7 β’ (πΉ(CircuitsβπΊ)π β (πΉ(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) | |
| 12 | 11 | simprd 494 | . . . . . 6 β’ (πΉ(CircuitsβπΊ)π β (πβ0) = (πβ(β―βπΉ))) |
| 13 | 12 | 3ad2ant3 1132 | . . . . 5 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β (πβ0) = (πβ(β―βπΉ))) |
| 14 | 13 | iftrued 4532 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))}) = β ) |
| 15 | 14 | eqeq2d 2736 | . . 3 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β ({π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))}) β {π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = β )) |
| 16 | rabeq0 4380 | . . . 4 β’ ({π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = β β βπ₯ β π Β¬ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)) | |
| 17 | notnotr 130 | . . . . 5 β’ (Β¬ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯) β 2 β₯ ((VtxDegβπΊ)βπ₯)) | |
| 18 | 17 | ralimi 3073 | . . . 4 β’ (βπ₯ β π Β¬ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) |
| 19 | 16, 18 | sylbi 216 | . . 3 β’ ({π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = β β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) |
| 20 | 15, 19 | biimtrdi 252 | . 2 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β ({π₯ β π β£ Β¬ 2 β₯ ((VtxDegβπΊ)βπ₯)} = if((πβ0) = (πβ(β―βπΉ)), β , {(πβ0), (πβ(β―βπΉ))}) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯))) |
| 21 | 10, 20 | mpd 15 | 1 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β βπ₯ β π 2 β₯ ((VtxDegβπΊ)βπ₯)) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3051 {crab 3419 β c0 4318 ifcif 4524 {cpr 4626 class class class wbr 5143 Fun wfun 6537 βcfv 6543 0cc0 11138 2c2 12297 β―chash 14321 β₯ cdvds 16230 Vtxcvtx 28853 iEdgciedg 28854 UHGraphcuhgr 28913 UPGraphcupgr 28937 VtxDegcvtxdg 29323 Trailsctrls 29548 Circuitsccrcts 29642 EulerPathsceupth 30051 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-inf 9466 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-rp 13007 df-xadd 13125 df-fz 13517 df-fzo 13660 df-seq 13999 df-exp 14059 df-hash 14322 df-word 14497 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-vtx 28855 df-iedg 28856 df-edg 28905 df-uhgr 28915 df-ushgr 28916 df-upgr 28939 df-uspgr 29007 df-vtxdg 29324 df-wlks 29457 df-trls 29550 df-crcts 29644 df-eupth 30052 |
| This theorem is referenced by: (None) |
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