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Mirrors > Home > MPE Home > Th. List > eulerpathpr | Structured version Visualization version GIF version |
Description: A graph 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 |
---|---|
eulerpathpr | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eulerpathpr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2726 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | simpl 481 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐺 ∈ UPGraph) | |
4 | upgruhgr 29035 | . . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph) | |
5 | 2 | uhgrfun 28999 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺)) |
7 | 6 | adantr 479 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → Fun (iEdg‘𝐺)) |
8 | simpr 483 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐹(EulerPaths‘𝐺)𝑃) | |
9 | 1, 2, 3, 7, 8 | eupth2 30169 | . . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → {𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})) |
10 | 9 | fveq2d 6897 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))) |
11 | fveq2 6893 | . . . 4 ⊢ (∅ = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}) → (♯‘∅) = (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))) | |
12 | 11 | eleq1d 2811 | . . 3 ⊢ (∅ = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}) → ((♯‘∅) ∈ {0, 2} ↔ (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})) ∈ {0, 2})) |
13 | fveq2 6893 | . . . 4 ⊢ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}) → (♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) = (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}))) | |
14 | 13 | eleq1d 2811 | . . 3 ⊢ ({(𝑃‘0), (𝑃‘(♯‘𝐹))} = if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))}) → ((♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) ∈ {0, 2} ↔ (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})) ∈ {0, 2})) |
15 | hash0 14379 | . . . . 5 ⊢ (♯‘∅) = 0 | |
16 | c0ex 11249 | . . . . . 6 ⊢ 0 ∈ V | |
17 | 16 | prid1 4761 | . . . . 5 ⊢ 0 ∈ {0, 2} |
18 | 15, 17 | eqeltri 2822 | . . . 4 ⊢ (♯‘∅) ∈ {0, 2} |
19 | 18 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (♯‘∅) ∈ {0, 2}) |
20 | simpr 483 | . . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) | |
21 | 20 | neqned 2937 | . . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) |
22 | fvex 6906 | . . . . . 6 ⊢ (𝑃‘0) ∈ V | |
23 | fvex 6906 | . . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) ∈ V | |
24 | hashprg 14407 | . . . . . 6 ⊢ (((𝑃‘0) ∈ V ∧ (𝑃‘(♯‘𝐹)) ∈ V) → ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) = 2)) | |
25 | 22, 23, 24 | mp2an 690 | . . . . 5 ⊢ ((𝑃‘0) ≠ (𝑃‘(♯‘𝐹)) ↔ (♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) = 2) |
26 | 21, 25 | sylib 217 | . . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) = 2) |
27 | 2ex 12335 | . . . . 5 ⊢ 2 ∈ V | |
28 | 27 | prid2 4762 | . . . 4 ⊢ 2 ∈ {0, 2} |
29 | 26, 28 | eqeltrdi 2834 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(♯‘𝐹))) → (♯‘{(𝑃‘0), (𝑃‘(♯‘𝐹))}) ∈ {0, 2}) |
30 | 12, 14, 19, 29 | ifbothda 4561 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (♯‘if((𝑃‘0) = (𝑃‘(♯‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(♯‘𝐹))})) ∈ {0, 2}) |
31 | 10, 30 | eqeltrd 2826 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (♯‘{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 ∈ wcel 2099 ≠ wne 2930 {crab 3419 Vcvv 3462 ∅c0 4322 ifcif 4523 {cpr 4625 class class class wbr 5145 Fun wfun 6540 ‘cfv 6546 0cc0 11149 2c2 12313 ♯chash 14342 ∥ cdvds 16251 Vtxcvtx 28929 iEdgciedg 28930 UHGraphcuhgr 28989 UPGraphcupgr 29013 VtxDegcvtxdg 29399 EulerPathsceupth 30127 |
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 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-inf 9479 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-xnn0 12591 df-z 12605 df-uz 12869 df-rp 13023 df-xadd 13141 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-word 14518 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-vtx 28931 df-iedg 28932 df-edg 28981 df-uhgr 28991 df-ushgr 28992 df-upgr 29015 df-uspgr 29083 df-vtxdg 29400 df-wlks 29533 df-trls 29626 df-eupth 30128 |
This theorem is referenced by: eulerpath 30171 |
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