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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 2726 | . . . 4 β’ (iEdgβπΊ) = (iEdgβπΊ) | |
| 3 | simpl 482 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΊ β UPGraph) | |
| 4 | upgruhgr 28870 | . . . . . 6 β’ (πΊ β UPGraph β πΊ β UHGraph) | |
| 5 | 2 | uhgrfun 28834 | . . . . . 6 β’ (πΊ β UHGraph β Fun (iEdgβπΊ)) |
| 6 | 4, 5 | syl 17 | . . . . 5 β’ (πΊ β UPGraph β Fun (iEdgβπΊ)) |
| 7 | 6 | adantr 480 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β Fun (iEdgβπΊ)) |
| 8 | simpr 484 | . . . 4 β’ ((πΊ β UPGraph β§ πΉ(EulerPathsβπΊ)π) β πΉ(EulerPathsβπΊ)π) | |
| 9 | 1, 2, 3, 7, 8 | eupth2 30001 | . . 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 29558 | . . . . . . 7 β’ (πΉ(CircuitsβπΊ)π β (πΉ(TrailsβπΊ)π β§ (πβ0) = (πβ(β―βπΉ)))) | |
| 12 | 11 | simprd 495 | . . . . . 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 4531 | . . . 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 2737 | . . 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 4379 | . . . 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 3077 | . . . 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 253 | . 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 β c0 4317 ifcif 4523 {cpr 4625 class class class wbr 5141 Fun wfun 6531 βcfv 6537 0cc0 11112 2c2 12271 β―chash 14295 β₯ cdvds 16204 Vtxcvtx 28764 iEdgciedg 28765 UHGraphcuhgr 28824 UPGraphcupgr 28848 VtxDegcvtxdg 29231 Trailsctrls 29456 Circuitsccrcts 29550 EulerPathsceupth 29959 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ifp 1060 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-word 14471 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16205 df-vtx 28766 df-iedg 28767 df-edg 28816 df-uhgr 28826 df-ushgr 28827 df-upgr 28850 df-uspgr 28918 df-vtxdg 29232 df-wlks 29365 df-trls 29458 df-crcts 29552 df-eupth 29960 |
| This theorem is referenced by: (None) |
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