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Theorem eulerpathum 16588

Description: A multigraph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 26-Feb-2021.)

**Hypothesis**
Ref	Expression
eulerpathpr.v	⊢ 𝑉 = (Vtx‘𝐺)

**Assertion**
Ref	Expression
eulerpathum	⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Distinct variable groups: 𝑥,𝐺 𝑥,𝑉 𝑗,𝐺 Allowed substitution hint: 𝑉(𝑗)

Proof of Theorem eulerpathum Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1025	. . 3 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺))
2		releupth 16551	. . . 4 ⊢ Rel (EulerPaths‘𝐺)
3		reldmm 4980	. . . 4 ⊢ (Rel (EulerPaths‘𝐺) → (∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ↔ ∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺)))
4	2, 3	ax-mp 5	. . 3 ⊢ (∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ↔ ∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺))
5	1, 4	sylib 122	. 2 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → ∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺))
6		simpr 110	. . . 4 ⊢ (((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) → 𝑓 ∈ dom (EulerPaths‘𝐺))
7		vex 2818	. . . . 5 ⊢ 𝑓 ∈ V
8	7	eldm 4958	. . . 4 ⊢ (𝑓 ∈ dom (EulerPaths‘𝐺) ↔ ∃𝑝 𝑓(EulerPaths‘𝐺)𝑝)
9	6, 8	sylib 122	. . 3 ⊢ (((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) → ∃𝑝 𝑓(EulerPaths‘𝐺)𝑝)
10		simpll1 1063	. . . 4 ⊢ ((((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) ∧ 𝑓(EulerPaths‘𝐺)𝑝) → 𝐺 ∈ UMGraph)
11		simpr 110	. . . 4 ⊢ ((((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) ∧ 𝑓(EulerPaths‘𝐺)𝑝) → 𝑓(EulerPaths‘𝐺)𝑝)
12		simpll3 1065	. . . 4 ⊢ ((((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) ∧ 𝑓(EulerPaths‘𝐺)𝑝) → 𝑉 ∈ Fin)
13		eulerpathpr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
14	13	eulerpathprum 16587	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(EulerPaths‘𝐺)𝑝 ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
15	10, 11, 12, 14	syl3anc 1274	. . 3 ⊢ ((((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) ∧ 𝑓(EulerPaths‘𝐺)𝑝) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
16	9, 15	exlimddv 1950	. 2 ⊢ (((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) ∧ 𝑓 ∈ dom (EulerPaths‘𝐺)) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
17	5, 16	exlimddv 1950	1 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∃wex 1541 ∈ wcel 2205 {crab 2526 {cpr 3695 class class class wbr 4114 dom cdm 4754 Rel wrel 4759 ‘cfv 5357 Fincfn 6988 0cc0 8143 2c2 9305 ♯chash 11163 ∥ cdvds 12498 Vtxcvtx 16119 UMGraphcumgr 16199 VtxDegcvtxdg 16393 EulerPathsceupth 16549

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-ifp 987 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-dec 9728 df-uz 9872 df-q 9970 df-rp 10005 df-xadd 10125 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-ihash 11164 df-word 11250 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499 df-ndx 13299 df-slot 13300 df-base 13302 df-edgf 16112 df-vtx 16121 df-iedg 16122 df-edg 16165 df-uhgrm 16176 df-ushgrm 16177 df-upgren 16200 df-umgren 16201 df-uspgren 16262 df-subgr 16361 df-vtxdg 16394 df-wlks 16425 df-trls 16488 df-eupth 16550

This theorem is referenced by: konigsberg 16600

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