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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 2825 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
3 | 1, 2 | iswlkg 26918 | . 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 4300 | . . . . 5 ⊢ ∀𝑘 ∈ ∅ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) | |
5 | hash0 13455 | . . . . . . . 8 ⊢ (♯‘∅) = 0 | |
6 | 5 | oveq2i 6921 | . . . . . . 7 ⊢ (0..^(♯‘∅)) = (0..^0) |
7 | fzo0 12794 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
8 | 6, 7 | eqtri 2849 | . . . . . 6 ⊢ (0..^(♯‘∅)) = ∅ |
9 | 8 | raleqi 3354 | . . . . 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 223 | . . . 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 525 | . . 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 2834 | . . . . . 6 ⊢ 0 = (♯‘∅) |
13 | 12 | oveq2i 6921 | . . . . 5 ⊢ (0...0) = (0...(♯‘∅)) |
14 | 13 | feq2i 6274 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:(0...(♯‘∅))⟶𝑉) |
15 | wrd0 13606 | . . . . 5 ⊢ ∅ ∈ Word dom (iEdg‘𝐺) | |
16 | 15 | biantrur 526 | . . . 4 ⊢ (𝑃:(0...(♯‘∅))⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉)) |
17 | 14, 16 | bitri 267 | . . 3 ⊢ (𝑃:(0...0)⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(♯‘∅))⟶𝑉)) |
18 | df-3an 1113 | . . 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 296 | . 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 | syl6bb 279 | 1 ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 if-wif 1089 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ⊆ wss 3798 ∅c0 4146 {csn 4399 {cpr 4401 class class class wbr 4875 dom cdm 5346 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 0cc0 10259 1c1 10260 + caddc 10262 ...cfz 12626 ..^cfzo 12767 ♯chash 13417 Word cword 13581 Vtxcvtx 26301 iEdgciedg 26302 Walkscwlks 26901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-ifp 1090 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-card 9085 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-fzo 12768 df-hash 13418 df-word 13582 df-wlks 26904 |
This theorem is referenced by: is0wlk 27489 0wlkon 27492 0trl 27494 0clwlk 27502 |
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