Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 01wlk | 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.) |
Ref | Expression |
---|---|
01wlk.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
01wlk | ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (∅(1Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
2 | 01wlk.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
4 | 2, 3 | is1wlk 40813 | . . 3 ⊢ ((𝐺 ∈ 𝑈 ∧ ∅ ∈ V ∧ 𝑃 ∈ 𝑍) → (∅(1Walks‘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)))))) |
5 | 1, 4 | mp3an2 1404 | . 2 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (∅(1Walks‘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)))))) |
6 | ral0 4028 | . . . . 5 ⊢ ∀𝑘 ∈ ∅ if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) | |
7 | hash0 13019 | . . . . . . . 8 ⊢ (#‘∅) = 0 | |
8 | 7 | oveq2i 6560 | . . . . . . 7 ⊢ (0..^(#‘∅)) = (0..^0) |
9 | fzo0 12361 | . . . . . . 7 ⊢ (0..^0) = ∅ | |
10 | 8, 9 | eqtri 2632 | . . . . . 6 ⊢ (0..^(#‘∅)) = ∅ |
11 | 10 | raleqi 3119 | . . . . 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)))) |
12 | 6, 11 | mpbir 220 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) |
13 | 12 | 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))))) |
14 | 7 | eqcomi 2619 | . . . . . 6 ⊢ 0 = (#‘∅) |
15 | 14 | oveq2i 6560 | . . . . 5 ⊢ (0...0) = (0...(#‘∅)) |
16 | 15 | feq2i 5950 | . . . 4 ⊢ (𝑃:(0...0)⟶𝑉 ↔ 𝑃:(0...(#‘∅))⟶𝑉) |
17 | wrd0 13185 | . . . . 5 ⊢ ∅ ∈ Word dom (iEdg‘𝐺) | |
18 | 17 | biantrur 526 | . . . 4 ⊢ (𝑃:(0...(#‘∅))⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉)) |
19 | 16, 18 | bitri 263 | . . 3 ⊢ (𝑃:(0...0)⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉)) |
20 | df-3an 1033 | . . 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))))) | |
21 | 13, 19, 20 | 3bitr4ri 292 | . 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) |
22 | 5, 21 | syl6bb 275 | 1 ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (∅(1Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 if-wif 1006 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 {csn 4125 {cpr 4127 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 Vtxcvtx 25673 iEdgciedg 25674 1Walksc1wlks 40796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-1wlks 40800 |
This theorem is referenced by: is01wlk 41285 0wlkOn 41288 0Trl 41290 0clWlk 41298 |
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