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Mirrors > Home > MPE Home > Th. List > eulerpath | Structured version Visualization version GIF version |
Description: A pseudograph 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.) |
Ref | Expression |
---|---|
eulerpathpr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
eulerpath | ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releupth 30132 | . . . . . 6 ⊢ Rel (EulerPaths‘𝐺) | |
2 | reldm0 5934 | . . . . . 6 ⊢ (Rel (EulerPaths‘𝐺) → ((EulerPaths‘𝐺) = ∅ ↔ dom (EulerPaths‘𝐺) = ∅)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ((EulerPaths‘𝐺) = ∅ ↔ dom (EulerPaths‘𝐺) = ∅) |
4 | 3 | necon3bii 2983 | . . . 4 ⊢ ((EulerPaths‘𝐺) ≠ ∅ ↔ dom (EulerPaths‘𝐺) ≠ ∅) |
5 | n0 4349 | . . . 4 ⊢ (dom (EulerPaths‘𝐺) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺)) | |
6 | 4, 5 | bitri 274 | . . 3 ⊢ ((EulerPaths‘𝐺) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺)) |
7 | vex 3466 | . . . . . 6 ⊢ 𝑓 ∈ V | |
8 | 7 | eldm 5907 | . . . . 5 ⊢ (𝑓 ∈ dom (EulerPaths‘𝐺) ↔ ∃𝑝 𝑓(EulerPaths‘𝐺)𝑝) |
9 | eulerpathpr.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | 9 | eulerpathpr 30173 | . . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑓(EulerPaths‘𝐺)𝑝) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
11 | 10 | expcom 412 | . . . . . 6 ⊢ (𝑓(EulerPaths‘𝐺)𝑝 → (𝐺 ∈ UPGraph → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})) |
12 | 11 | exlimiv 1926 | . . . . 5 ⊢ (∃𝑝 𝑓(EulerPaths‘𝐺)𝑝 → (𝐺 ∈ UPGraph → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})) |
13 | 8, 12 | sylbi 216 | . . . 4 ⊢ (𝑓 ∈ dom (EulerPaths‘𝐺) → (𝐺 ∈ UPGraph → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})) |
14 | 13 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓 ∈ dom (EulerPaths‘𝐺) → (𝐺 ∈ UPGraph → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})) |
15 | 6, 14 | sylbi 216 | . 2 ⊢ ((EulerPaths‘𝐺) ≠ ∅ → (𝐺 ∈ UPGraph → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})) |
16 | 15 | impcom 406 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 {crab 3419 ∅c0 4325 {cpr 4635 class class class wbr 5153 dom cdm 5682 Rel wrel 5687 ‘cfv 6554 0cc0 11158 2c2 12319 ♯chash 14347 ∥ cdvds 16256 Vtxcvtx 28932 UPGraphcupgr 29016 VtxDegcvtxdg 29402 EulerPathsceupth 30130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-map 8857 df-pm 8858 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-sup 9485 df-inf 9486 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12597 df-z 12611 df-uz 12875 df-rp 13029 df-xadd 13147 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-word 14523 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-dvds 16257 df-vtx 28934 df-iedg 28935 df-edg 28984 df-uhgr 28994 df-ushgr 28995 df-upgr 29018 df-uspgr 29086 df-vtxdg 29403 df-wlks 29536 df-trls 29629 df-eupth 30131 |
This theorem is referenced by: konigsberg 30190 |
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