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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 4876 | . . . . 5 ⊢ 〈{𝑋, 𝑌}, 𝐼〉 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
4 | 1, 3 | eqtri 2760 | . . . 4 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
5 | wlk2v2e.x | . . . . 5 ⊢ 𝑋 ∈ V | |
6 | wlk2v2e.y | . . . . 5 ⊢ 𝑌 ∈ V | |
7 | uspgr2v1e2w 28497 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph) | |
8 | 5, 6, 7 | mp2an 690 | . . . 4 ⊢ 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph |
9 | 4, 8 | eqeltri 2829 | . . 3 ⊢ 𝐺 ∈ USPGraph |
10 | uspgrupgr 28425 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ 𝐺 ∈ UPGraph |
12 | wlk2v2e.f | . . . . 5 ⊢ 𝐹 = 〈“00”〉 | |
13 | 2, 12 | wlk2v2elem1 29397 | . . . 4 ⊢ 𝐹 ∈ Word dom 𝐼 |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 5 | prid1 4765 | . . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋, 𝑌} |
16 | 6 | prid2 4766 | . . . . . . . . 9 ⊢ 𝑌 ∈ {𝑋, 𝑌} |
17 | s3cl 14826 | . . . . . . . . 9 ⊢ ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌}) | |
18 | 15, 16, 15, 17 | mp3an 1461 | . . . . . . . 8 ⊢ 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌} |
19 | 14, 18 | eqeltri 2829 | . . . . . . 7 ⊢ 𝑃 ∈ Word {𝑋, 𝑌} |
20 | wrdf 14465 | . . . . . . 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 6891 | . . . . . . . . 9 ⊢ (♯‘𝑃) = (♯‘〈“𝑋𝑌𝑋”〉) |
23 | s3len 14841 | . . . . . . . . 9 ⊢ (♯‘〈“𝑋𝑌𝑋”〉) = 3 | |
24 | 22, 23 | eqtr2i 2761 | . . . . . . . 8 ⊢ 3 = (♯‘𝑃) |
25 | 24 | oveq2i 7416 | . . . . . . 7 ⊢ (0..^3) = (0..^(♯‘𝑃)) |
26 | 25 | feq2i 6706 | . . . . . 6 ⊢ (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(♯‘𝑃))⟶{𝑋, 𝑌}) |
27 | 21, 26 | mpbir 230 | . . . . 5 ⊢ 𝑃:(0..^3)⟶{𝑋, 𝑌} |
28 | 12 | fveq2i 6891 | . . . . . . . . 9 ⊢ (♯‘𝐹) = (♯‘〈“00”〉) |
29 | s2len 14836 | . . . . . . . . 9 ⊢ (♯‘〈“00”〉) = 2 | |
30 | 28, 29 | eqtri 2760 | . . . . . . . 8 ⊢ (♯‘𝐹) = 2 |
31 | 30 | oveq2i 7416 | . . . . . . 7 ⊢ (0...(♯‘𝐹)) = (0...2) |
32 | 3z 12591 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
33 | fzoval 13629 | . . . . . . . . 9 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
34 | 32, 33 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^3) = (0...(3 − 1)) |
35 | 3m1e2 12336 | . . . . . . . . 9 ⊢ (3 − 1) = 2 | |
36 | 35 | oveq2i 7416 | . . . . . . . 8 ⊢ (0...(3 − 1)) = (0...2) |
37 | 34, 36 | eqtr2i 2761 | . . . . . . 7 ⊢ (0...2) = (0..^3) |
38 | 31, 37 | eqtri 2760 | . . . . . 6 ⊢ (0...(♯‘𝐹)) = (0..^3) |
39 | 38 | feq2i 6706 | . . . . 5 ⊢ (𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌}) |
40 | 27, 39 | mpbir 230 | . . . 4 ⊢ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} |
41 | 2, 12, 5, 6, 14 | wlk2v2elem2 29398 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
42 | 13, 40, 41 | 3pm3.2i 1339 | . . 3 ⊢ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
43 | 1 | fveq2i 6891 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) |
44 | prex 5431 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
45 | s1cli 14551 | . . . . . . 7 ⊢ 〈“{𝑋, 𝑌}”〉 ∈ Word V | |
46 | 2, 45 | eqeltri 2829 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
47 | opvtxfv 28253 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌}) | |
48 | 44, 46, 47 | mp2an 690 | . . . . 5 ⊢ (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌} |
49 | 43, 48 | eqtr2i 2761 | . . . 4 ⊢ {𝑋, 𝑌} = (Vtx‘𝐺) |
50 | 1 | fveq2i 6891 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) |
51 | opiedgfv 28256 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼) | |
52 | 44, 46, 51 | mp2an 690 | . . . . 5 ⊢ (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼 |
53 | 50, 52 | eqtr2i 2761 | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) |
54 | 49, 53 | upgriswlk 28887 | . . 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 257 | . 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 {cpr 4629 〈cop 4633 class class class wbr 5147 dom cdm 5675 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 − cmin 11440 2c2 12263 3c3 12264 ℤcz 12554 ...cfz 13480 ..^cfzo 13623 ♯chash 14286 Word cword 14460 〈“cs1 14541 〈“cs2 14788 〈“cs3 14789 Vtxcvtx 28245 iEdgciedg 28246 UPGraphcupgr 28329 USPGraphcuspgr 28397 Walkscwlks 28842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-vtx 28247 df-iedg 28248 df-edg 28297 df-uhgr 28307 df-upgr 28331 df-uspgr 28399 df-wlks 28845 |
This theorem is referenced by: (None) |
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