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Theorem eulerpathum 16402

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 16365	. . . 4 ⊢ Rel (EulerPaths‘𝐺)
3		reldmm 4956	. . . 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 2806	. . . . 5 ⊢ 𝑓 ∈ V
8	7	eldm 4934	. . . 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 16401	. . . 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 1947	. 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 1947	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 2202 {crab 2515 {cpr 3674 class class class wbr 4093 dom cdm 4731 Rel wrel 4736 ‘cfv 5333 Fincfn 6952 0cc0 8075 2c2 9237 ♯chash 11081 ∥ cdvds 12409 Vtxcvtx 15933 UMGraphcumgr 16013 VtxDegcvtxdg 16207 EulerPathsceupth 16363

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-map 6862 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-7 9250 df-8 9251 df-9 9252 df-n0 9446 df-z 9523 df-dec 9655 df-uz 9799 df-q 9897 df-rp 9932 df-xadd 10051 df-fz 10287 df-fzo 10421 df-fl 10574 df-mod 10629 df-seqfrec 10754 df-exp 10845 df-ihash 11082 df-word 11161 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620 df-dvds 12410 df-ndx 13146 df-slot 13147 df-base 13149 df-edgf 15926 df-vtx 15935 df-iedg 15936 df-edg 15979 df-uhgrm 15990 df-ushgrm 15991 df-upgren 16014 df-umgren 16015 df-uspgren 16076 df-subgr 16175 df-vtxdg 16208 df-wlks 16239 df-trls 16302 df-eupth 16364

This theorem is referenced by: konigsberg 16414

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