![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2799 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈0, 𝐴〉} = {〈0, 𝐴〉}) | |
2 | eupth0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | is0wlk 27902 | . . . 4 ⊢ (({〈0, 𝐴〉} = {〈0, 𝐴〉} ∧ 𝐴 ∈ 𝑉) → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
4 | 1, 3 | mpancom 687 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
5 | f1o0 6626 | . . . 4 ⊢ ∅:∅–1-1-onto→∅ | |
6 | eqidd 2799 | . . . . 5 ⊢ (𝐼 = ∅ → ∅ = ∅) | |
7 | hash0 13724 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
8 | 7 | oveq2i 7146 | . . . . . . 7 ⊢ (0..^(♯‘∅)) = (0..^0) |
9 | fzo0 13056 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtri 2821 | . . . . . 6 ⊢ (0..^(♯‘∅)) = ∅ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐼 = ∅ → (0..^(♯‘∅)) = ∅) |
12 | dmeq 5736 | . . . . . 6 ⊢ (𝐼 = ∅ → dom 𝐼 = dom ∅) | |
13 | dm0 5754 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | eqtrdi 2849 | . . . . 5 ⊢ (𝐼 = ∅ → dom 𝐼 = ∅) |
15 | 6, 11, 14 | f1oeq123d 6585 | . . . 4 ⊢ (𝐼 = ∅ → (∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼 ↔ ∅:∅–1-1-onto→∅)) |
16 | 5, 15 | mpbiri 261 | . . 3 ⊢ (𝐼 = ∅ → ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼) |
17 | 4, 16 | anim12i 615 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼)) |
18 | eupth0.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
19 | 18 | iseupthf1o 27987 | . 2 ⊢ (∅(EulerPaths‘𝐺){〈0, 𝐴〉} ↔ (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼)) |
20 | 17, 19 | sylibr 237 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∅c0 4243 {csn 4525 〈cop 4531 class class class wbr 5030 dom cdm 5519 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 ..^cfzo 13028 ♯chash 13686 Vtxcvtx 26789 iEdgciedg 26790 Walkscwlks 27386 EulerPathsceupth 27982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ifp 1059 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-wlks 27389 df-trls 27482 df-eupth 27983 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |