![]() |
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 2736 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈0, 𝐴〉} = {〈0, 𝐴〉}) | |
2 | eupth0.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 2 | is0wlk 30146 | . . . 4 ⊢ (({〈0, 𝐴〉} = {〈0, 𝐴〉} ∧ 𝐴 ∈ 𝑉) → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
4 | 1, 3 | mpancom 688 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∅(Walks‘𝐺){〈0, 𝐴〉}) |
5 | f1o0 6886 | . . . 4 ⊢ ∅:∅–1-1-onto→∅ | |
6 | eqidd 2736 | . . . . 5 ⊢ (𝐼 = ∅ → ∅ = ∅) | |
7 | hash0 14403 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
8 | 7 | oveq2i 7442 | . . . . . . 7 ⊢ (0..^(♯‘∅)) = (0..^0) |
9 | fzo0 13720 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtri 2763 | . . . . . 6 ⊢ (0..^(♯‘∅)) = ∅ |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐼 = ∅ → (0..^(♯‘∅)) = ∅) |
12 | dmeq 5917 | . . . . . 6 ⊢ (𝐼 = ∅ → dom 𝐼 = dom ∅) | |
13 | dm0 5934 | . . . . . 6 ⊢ dom ∅ = ∅ | |
14 | 12, 13 | eqtrdi 2791 | . . . . 5 ⊢ (𝐼 = ∅ → dom 𝐼 = ∅) |
15 | 6, 11, 14 | f1oeq123d 6843 | . . . 4 ⊢ (𝐼 = ∅ → (∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼 ↔ ∅:∅–1-1-onto→∅)) |
16 | 5, 15 | mpbiri 258 | . . 3 ⊢ (𝐼 = ∅ → ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼) |
17 | 4, 16 | anim12i 613 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼)) |
18 | eupth0.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
19 | 18 | iseupthf1o 30231 | . 2 ⊢ (∅(EulerPaths‘𝐺){〈0, 𝐴〉} ↔ (∅(Walks‘𝐺){〈0, 𝐴〉} ∧ ∅:(0..^(♯‘∅))–1-1-onto→dom 𝐼)) |
20 | 17, 19 | sylibr 234 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 = ∅) → ∅(EulerPaths‘𝐺){〈0, 𝐴〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {csn 4631 〈cop 4637 class class class wbr 5148 dom cdm 5689 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ..^cfzo 13691 ♯chash 14366 Vtxcvtx 29028 iEdgciedg 29029 Walkscwlks 29629 EulerPathsceupth 30226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-wlks 29632 df-trls 29725 df-eupth 30227 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |