eulerpathum - Intuitionistic Logic Explorer

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Theorem eulerpathum 16893

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 1029	. . 3 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺))
2		releupth 16856	. . . 4 ⊢ Rel (EulerPaths‘𝐺)
3		reldmm 5000	. . . 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 2824	. . . . 5 ⊢ 𝑓 ∈ V
8	7	eldm 4978	. . . 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 1067	. . . 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 1069	. . . 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 16892	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ 𝑓(EulerPaths‘𝐺)𝑝 ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
15	10, 11, 12, 14	syl3anc 1278	. . 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 1954	. 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 1954	1 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {crab 2532 {cpr 3710 class class class wbr 4130 dom cdm 4774 Rel wrel 4779 ‘cfv 5377 Fincfn 7022 0cc0 8180 2c2 9358 ♯chash 11230 ∥ cdvds 12573 Vtxcvtx 16424 UMGraphcumgr 16504 VtxDegcvtxdg 16698 EulerPathsceupth 16854

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8271 ax-resscn 8272 ax-1cn 8273 ax-1re 8274 ax-icn 8275 ax-addcl 8276 ax-addrcl 8277 ax-mulcl 8278 ax-mulrcl 8279 ax-addcom 8280 ax-mulcom 8281 ax-addass 8282 ax-mulass 8283 ax-distr 8284 ax-i2m1 8285 ax-0lt1 8286 ax-1rid 8287 ax-0id 8288 ax-rnegex 8289 ax-precex 8290 ax-cnre 8291 ax-pre-ltirr 8292 ax-pre-ltwlin 8293 ax-pre-lttrn 8294 ax-pre-apti 8295 ax-pre-ltadd 8296 ax-pre-mulgt0 8297 ax-pre-mulext 8298 ax-arch 8299

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-ifp 991 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8363 df-mnf 8364 df-xr 8365 df-ltxr 8366 df-le 8367 df-sub 8501 df-neg 8502 df-reap 8906 df-ap 8913 df-div 9006 df-inn 9308 df-2 9366 df-3 9367 df-4 9368 df-5 9369 df-6 9370 df-7 9371 df-8 9372 df-9 9373 df-n0 9569 df-z 9650 df-dec 9783 df-uz 9932 df-q 10030 df-rp 10066 df-xadd 10186 df-fz 10423 df-fzo 10561 df-fl 10716 df-mod 10775 df-seqfrec 10900 df-exp 10991 df-ihash 11231 df-word 11321 df-cj 11623 df-re 11624 df-im 11625 df-rsqrt 11780 df-abs 11781 df-dvds 12574 df-ndx 13407 df-slot 13408 df-base 13410 df-edgf 16417 df-vtx 16426 df-iedg 16427 df-edg 16470 df-uhgrm 16481 df-ushgrm 16482 df-upgren 16505 df-umgren 16506 df-uspgren 16567 df-subgr 16666 df-vtxdg 16699 df-wlks 16730 df-trls 16793 df-eupth 16855

This theorem is used by: konigsberg 16905

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