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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 2622 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | simpl 473 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐺 ∈ UPGraph ) | |
4 | upgruhgr 25997 | . . . . . 6 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph ) | |
5 | 2 | uhgrfun 25961 | . . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺)) |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → Fun (iEdg‘𝐺)) |
8 | simpr 477 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → 𝐹(EulerPaths‘𝐺)𝑃) | |
9 | 1, 2, 3, 7, 8 | eupth2 27099 | . . 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 6195 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))) |
11 | fveq2 6191 | . . . 4 ⊢ (∅ = if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}) → (#‘∅) = (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))}))) | |
12 | 11 | eleq1d 2686 | . . 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 6191 | . . . 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 2686 | . . 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 13158 | . . . . 5 ⊢ (#‘∅) = 0 | |
16 | c0ex 10034 | . . . . . 6 ⊢ 0 ∈ V | |
17 | 16 | prid1 4297 | . . . . 5 ⊢ 0 ∈ {0, 2} |
18 | 15, 17 | eqeltri 2697 | . . . 4 ⊢ (#‘∅) ∈ {0, 2} |
19 | 18 | a1i 11 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘∅) ∈ {0, 2}) |
20 | simpr 477 | . . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) | |
21 | 20 | neqned 2801 | . . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) |
22 | fvex 6201 | . . . . . 6 ⊢ (𝑃‘0) ∈ V | |
23 | fvex 6201 | . . . . . 6 ⊢ (𝑃‘(#‘𝐹)) ∈ V | |
24 | hashprg 13182 | . . . . . 6 ⊢ (((𝑃‘0) ∈ V ∧ (𝑃‘(#‘𝐹)) ∈ V) → ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2)) | |
25 | 22, 23, 24 | mp2an 708 | . . . . 5 ⊢ ((𝑃‘0) ≠ (𝑃‘(#‘𝐹)) ↔ (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2) |
26 | 21, 25 | sylib 208 | . . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) = 2) |
27 | 2ex 11092 | . . . . 5 ⊢ 2 ∈ V | |
28 | 27 | prid2 4298 | . . . 4 ⊢ 2 ∈ {0, 2} |
29 | 26, 28 | syl6eqel 2709 | . . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) ∧ ¬ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (#‘{(𝑃‘0), (𝑃‘(#‘𝐹))}) ∈ {0, 2}) |
30 | 12, 14, 19, 29 | ifbothda 4123 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(EulerPaths‘𝐺)𝑃) → (#‘if((𝑃‘0) = (𝑃‘(#‘𝐹)), ∅, {(𝑃‘0), (𝑃‘(#‘𝐹))})) ∈ {0, 2}) |
31 | 10, 30 | eqeltrd 2701 | 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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 {crab 2916 Vcvv 3200 ∅c0 3915 ifcif 4086 {cpr 4179 class class class wbr 4653 Fun wfun 5882 ‘cfv 5888 0cc0 9936 2c2 11070 #chash 13117 ∥ cdvds 14983 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 UPGraph cupgr 25975 VtxDegcvtxdg 26361 EulerPathsceupth 27057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-xadd 11947 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-word 13299 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-ushgr 25954 df-upgr 25977 df-uspgr 26045 df-vtxdg 26362 df-wlks 26495 df-trls 26589 df-eupth 27058 |
This theorem is referenced by: eulerpath 27101 |
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