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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 2732 | . . . 4 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 3 | simpl 483 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΊ β UPGraph) | |
| 4 | upgruhgr 28351 | . . . . . 6 β’ (πΊ β UPGraph β πΊ β UHGraph) | |
| 5 | 2 | uhgrfun 28315 | . . . . . 6 β’ (πΊ β UHGraph β Fun (iEdgβπΊ)) |
| 6 | 4, 5 | syl 17 | . . . . 5 β’ (πΊ β UPGraph β Fun (iEdgβπΊ)) |
| 7 | 6 | adantr 481 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β Fun (iEdgβπΊ)) |
| 8 | simpr 485 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΉ(EulerPathsβπΊ)π) | |
| 9 | 1, 2, 3, 7, 8 | eupth2 29481 | . . 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 1132 | . 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 29038 | . . . . . . 7 β’ (πΉ(CircuitsβπΊ)π β (πΉ(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) | |
| 12 | 11 | simprd 496 | . . . . . 6 β’ (πΉ(CircuitsβπΊ)π β (πβ0) = (πβ(β―βπΉ))) |
| 13 | 12 | 3ad2ant3 1135 | . . . . 5 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π β§ πΉ(CircuitsβπΊ)π) β (πβ0) = (πβ(β―βπΉ))) |
| 14 | 13 | iftrued 4535 | . . . 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 2743 | . . 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 4383 | . . . 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 3083 | . . . 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 | syl6bi 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 {crab 3432 β c0 4321 ifcif 4527 {cpr 4629 class class class wbr 5147 Fun wfun 6534 βcfv 6540 0cc0 11106 2c2 12263 β―chash 14286 β₯ cdvds 16193 Vtxcvtx 28245 iEdgciedg 28246 UHGraphcuhgr 28305 UPGraphcupgr 28329 VtxDegcvtxdg 28711 Trailsctrls 28936 Circuitsccrcts 29030 EulerPathsceupth 29439 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-rp 12971 df-xadd 13089 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-word 14461 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-vtx 28247 df-iedg 28248 df-edg 28297 df-uhgr 28307 df-ushgr 28308 df-upgr 28331 df-uspgr 28399 df-vtxdg 28712 df-wlks 28845 df-trls 28938 df-crcts 29032 df-eupth 29440 |
| This theorem is referenced by: (None) |
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