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Mirrors > Home > MPE Home > Th. List > 0wlk | Structured version Visualization version GIF version |
Description: A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
0wlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
0wlk | ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0wlk.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | eqid 2738 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | iswlkg 27555 | . 2 ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))))) |
4 | ral0 4400 | . . . . 5 ⊢ ∀𝑘 ∈ ∅ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) | |
5 | hash0 13821 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
6 | 5 | oveq2i 7182 | . . . . . . 7 ⊢ (0..^(♯‘∅)) = (0..^0) |
7 | fzo0 13153 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
8 | 6, 7 | eqtri 2761 | . . . . . 6 ⊢ (0..^(♯‘∅)) = ∅ |
9 | 8 | raleqi 3314 | . . . . 5 ⊢ (∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) ↔ ∀𝑘 ∈ ∅ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))) |
10 | 4, 9 | mpbir 234 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) |
11 | 10 | biantru 533 | . . 3 ⊢ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉) ↔ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))))) |
12 | 5 | eqcomi 2747 | . . . . . 6 ⊢ 0 = (♯‘∅) |
13 | 12 | oveq2i 7182 | . . . . 5 ⊢ (0...0) = (0...(♯‘∅)) |
14 | 13 | feq2i 6497 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:(0...(♯‘∅))⟶𝑉) |
15 | wrd0 13981 | . . . . 5 ⊢ ∅ ∈ Word dom (iEdg‘𝐺) | |
16 | 15 | biantrur 534 | . . . 4 ⊢ (𝑃:(0...(♯‘∅))⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉)) |
17 | 14, 16 | bitri 278 | . . 3 ⊢ (𝑃:(0...0)⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉)) |
18 | df-3an 1090 | . . 3 ⊢ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))) ↔ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))))) | |
19 | 11, 17, 18 | 3bitr4ri 307 | . 2 ⊢ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))) ↔ 𝑃:(0...0)⟶𝑉) |
20 | 3, 19 | bitrdi 290 | 1 ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 if-wif 1062 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ∀wral 3053 ⊆ wss 3844 ∅c0 4212 {csn 4517 {cpr 4519 class class class wbr 5031 dom cdm 5526 ⟶wf 6336 ‘cfv 6340 (class class class)co 7171 0cc0 10616 1c1 10617 + caddc 10619 ...cfz 12982 ..^cfzo 13125 ♯chash 13783 Word cword 13956 Vtxcvtx 26941 iEdgciedg 26942 Walkscwlks 27538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-1o 8132 df-er 8321 df-map 8440 df-pm 8441 df-en 8557 df-dom 8558 df-sdom 8559 df-fin 8560 df-card 9442 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-nn 11718 df-n0 11978 df-z 12064 df-uz 12326 df-fz 12983 df-fzo 13126 df-hash 13784 df-word 13957 df-wlks 27541 |
This theorem is referenced by: is0wlk 28054 0wlkon 28057 0trl 28059 0clwlk 28067 |
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