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Mirrors > Home > MPE Home > Th. List > wlk2v2e | Structured version Visualization version GIF version |
Description: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋 ≠ 𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
Ref | Expression |
---|---|
wlk2v2e.i | ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 |
wlk2v2e.f | ⊢ 𝐹 = 〈“00”〉 |
wlk2v2e.x | ⊢ 𝑋 ∈ V |
wlk2v2e.y | ⊢ 𝑌 ∈ V |
wlk2v2e.p | ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 |
wlk2v2e.g | ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 |
Ref | Expression |
---|---|
wlk2v2e | ⊢ 𝐹(Walks‘𝐺)𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wlk2v2e.g | . . . . 5 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 𝐼〉 | |
2 | wlk2v2e.i | . . . . . 6 ⊢ 𝐼 = 〈“{𝑋, 𝑌}”〉 | |
3 | 2 | opeq2i 4799 | . . . . 5 ⊢ 〈{𝑋, 𝑌}, 𝐼〉 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
4 | 1, 3 | eqtri 2841 | . . . 4 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
5 | wlk2v2e.x | . . . . 5 ⊢ 𝑋 ∈ V | |
6 | wlk2v2e.y | . . . . 5 ⊢ 𝑌 ∈ V | |
7 | uspgr2v1e2w 26960 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph) | |
8 | 5, 6, 7 | mp2an 688 | . . . 4 ⊢ 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph |
9 | 4, 8 | eqeltri 2906 | . . 3 ⊢ 𝐺 ∈ USPGraph |
10 | uspgrupgr 26888 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ 𝐺 ∈ UPGraph |
12 | wlk2v2e.f | . . . . 5 ⊢ 𝐹 = 〈“00”〉 | |
13 | 2, 12 | wlk2v2elem1 27861 | . . . 4 ⊢ 𝐹 ∈ Word dom 𝐼 |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 5 | prid1 4690 | . . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋, 𝑌} |
16 | 6 | prid2 4691 | . . . . . . . . 9 ⊢ 𝑌 ∈ {𝑋, 𝑌} |
17 | s3cl 14229 | . . . . . . . . 9 ⊢ ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌}) | |
18 | 15, 16, 15, 17 | mp3an 1452 | . . . . . . . 8 ⊢ 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌} |
19 | 14, 18 | eqeltri 2906 | . . . . . . 7 ⊢ 𝑃 ∈ Word {𝑋, 𝑌} |
20 | wrdf 13854 | . . . . . . 7 ⊢ (𝑃 ∈ Word {𝑋, 𝑌} → 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌}) | |
21 | 19, 20 | ax-mp 5 | . . . . . 6 ⊢ 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌} |
22 | 14 | fveq2i 6666 | . . . . . . . . 9 ⊢ (♯‘𝑃) = (♯‘〈“𝑋𝑌𝑋”〉) |
23 | s3len 14244 | . . . . . . . . 9 ⊢ (♯‘〈“𝑋𝑌𝑋”〉) = 3 | |
24 | 22, 23 | eqtr2i 2842 | . . . . . . . 8 ⊢ 3 = (♯‘𝑃) |
25 | 24 | oveq2i 7156 | . . . . . . 7 ⊢ (0..^3) = (0..^(♯‘𝑃)) |
26 | 25 | feq2i 6499 | . . . . . 6 ⊢ (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌}) |
27 | 21, 26 | mpbir 232 | . . . . 5 ⊢ 𝑃:(0..^3)⟶{𝑋, 𝑌} |
28 | 12 | fveq2i 6666 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“00”〉) |
29 | s2len 14239 | . . . . . . . . 9 ⊢ (♯‘〈“00”〉) = 2 | |
30 | 28, 29 | eqtri 2841 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
31 | 30 | oveq2i 7156 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) = (0...2) |
32 | 3z 12003 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
33 | fzoval 13027 | . . . . . . . . 9 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
34 | 32, 33 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^3) = (0...(3 − 1)) |
35 | 3m1e2 11753 | . . . . . . . . 9 ⊢ (3 − 1) = 2 | |
36 | 35 | oveq2i 7156 | . . . . . . . 8 ⊢ (0...(3 − 1)) = (0...2) |
37 | 34, 36 | eqtr2i 2842 | . . . . . . 7 ⊢ (0...2) = (0..^3) |
38 | 31, 37 | eqtri 2841 | . . . . . 6 ⊢ (0...(♯‘𝐹)) = (0..^3) |
39 | 38 | feq2i 6499 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌}) |
40 | 27, 39 | mpbir 232 | . . . 4 ⊢ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} |
41 | 2, 12, 5, 6, 14 | wlk2v2elem2 27862 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
42 | 13, 40, 41 | 3pm3.2i 1331 | . . 3 ⊢ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
43 | 1 | fveq2i 6666 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) |
44 | prex 5323 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
45 | s1cli 13947 | . . . . . . 7 ⊢ 〈“{𝑋, 𝑌}”〉 ∈ Word V | |
46 | 2, 45 | eqeltri 2906 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
47 | opvtxfv 26716 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌}) | |
48 | 44, 46, 47 | mp2an 688 | . . . . 5 ⊢ (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌} |
49 | 43, 48 | eqtr2i 2842 | . . . 4 ⊢ {𝑋, 𝑌} = (Vtx‘𝐺) |
50 | 1 | fveq2i 6666 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) |
51 | opiedgfv 26719 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼) | |
52 | 44, 46, 51 | mp2an 688 | . . . . 5 ⊢ (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼 |
53 | 50, 52 | eqtr2i 2842 | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) |
54 | 49, 53 | upgriswlk 27349 | . . 3 ⊢ (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
55 | 42, 54 | mpbiri 259 | . 2 ⊢ (𝐺 ∈ UPGraph → 𝐹(Walks‘𝐺)𝑃) |
56 | 11, 55 | ax-mp 5 | 1 ⊢ 𝐹(Walks‘𝐺)𝑃 |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 {cpr 4559 〈cop 4563 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 − cmin 10858 2c2 11680 3c3 11681 ℤcz 11969 ...cfz 12880 ..^cfzo 13021 ♯chash 13678 Word cword 13849 〈“cs1 13937 〈“cs2 14191 〈“cs3 14192 Vtxcvtx 26708 iEdgciedg 26709 UPGraphcupgr 26792 USPGraphcuspgr 26860 Walkscwlks 27305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-vtx 26710 df-iedg 26711 df-edg 26760 df-uhgr 26770 df-upgr 26794 df-uspgr 26862 df-wlks 27308 |
This theorem is referenced by: (None) |
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