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Mirrors > Home > MPE Home > Th. List > eupth0 | Structured version Visualization version GIF version |
Description: There is an Eulerian path on an empty graph, i.e. a graph with at least one vertex, but without an edge. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 5-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
eupth0.v | ⊢ 𝑉 = (Vtx‘𝐺) |
eupth0.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
eupth0 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈0, 𝐴〉} = {〈0, 𝐴〉}) | |
2 | eupth0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | is0wlk 26978 | . . . 4 ⊢ (({〈0, 𝐴〉} = {〈0, 𝐴〉} ∧ 𝐴 ∈ 𝑉) → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
4 | 1, 3 | mpancom 703 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
5 | f1o0 6173 | . . . 4 ⊢ ∅:∅–1-1-onto→∅ | |
6 | eqidd 2623 | . . . . 5 ⊢ (𝐼 = ∅ → ∅ = ∅) | |
7 | hash0 13158 | . . . . . . . 8 ⊢ (#‘∅) = 0 | |
8 | 7 | oveq2i 6661 | . . . . . . 7 ⊢ (0..^(#‘∅)) = (0..^0) |
9 | fzo0 12492 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtri 2644 | . . . . . 6 ⊢ (0..^(#‘∅)) = ∅ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐼 = ∅ → (0..^(#‘∅)) = ∅) |
12 | dmeq 5324 | . . . . . 6 ⊢ (𝐼 = ∅ → dom 𝐼 = dom ∅) | |
13 | dm0 5339 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | syl6eq 2672 | . . . . 5 ⊢ (𝐼 = ∅ → dom 𝐼 = ∅) |
15 | 6, 11, 14 | f1oeq123d 6133 | . . . 4 ⊢ (𝐼 = ∅ → (∅:(0..^(#‘∅))–1-1-onto→dom 𝐼 ↔ ∅:∅–1-1-onto→∅)) |
16 | 5, 15 | mpbiri 248 | . . 3 ⊢ (𝐼 = ∅ → ∅:(0..^(#‘∅))–1-1-onto→dom 𝐼) |
17 | 4, 16 | anim12i 590 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(#‘∅))–1-1-onto→dom 𝐼)) |
18 | eupth0.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
19 | 18 | iseupthf1o 27062 | . 2 ⊢ (∅(EulerPaths‘𝐺){〈0, 𝐴〉} ↔ (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(#‘∅))–1-1-onto→dom 𝐼)) |
20 | 17, 19 | sylibr 224 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∅c0 3915 {csn 4177 〈cop 4183 class class class wbr 4653 dom cdm 5114 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ..^cfzo 12465 #chash 13117 Vtxcvtx 25874 iEdgciedg 25875 Walkscwlks 26492 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 |
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-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-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-trls 26589 df-eupth 27058 |
This theorem is referenced by: (None) |
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