Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4406 | . . . . 5 ⊢ 〈{𝑋, 𝑌}, 𝐼〉 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
4 | 1, 3 | eqtri 2644 | . . . 4 ⊢ 𝐺 = 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 |
5 | wlk2v2e.x | . . . . 5 ⊢ 𝑋 ∈ V | |
6 | wlk2v2e.y | . . . . 5 ⊢ 𝑌 ∈ V | |
7 | uspgr2v1e2w 26143 | . . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph ) | |
8 | 5, 6, 7 | mp2an 708 | . . . 4 ⊢ 〈{𝑋, 𝑌}, 〈“{𝑋, 𝑌}”〉〉 ∈ USPGraph |
9 | 4, 8 | eqeltri 2697 | . . 3 ⊢ 𝐺 ∈ USPGraph |
10 | uspgrupgr 26071 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph ) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ 𝐺 ∈ UPGraph |
12 | wlk2v2e.f | . . . . 5 ⊢ 𝐹 = 〈“00”〉 | |
13 | 2, 12 | wlk2v2elem1 27015 | . . . 4 ⊢ 𝐹 ∈ Word dom 𝐼 |
14 | wlk2v2e.p | . . . . . . . 8 ⊢ 𝑃 = 〈“𝑋𝑌𝑋”〉 | |
15 | 5 | prid1 4297 | . . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋, 𝑌} |
16 | 6 | prid2 4298 | . . . . . . . . 9 ⊢ 𝑌 ∈ {𝑋, 𝑌} |
17 | s3cl 13624 | . . . . . . . . 9 ⊢ ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌}) | |
18 | 15, 16, 15, 17 | mp3an 1424 | . . . . . . . 8 ⊢ 〈“𝑋𝑌𝑋”〉 ∈ Word {𝑋, 𝑌} |
19 | 14, 18 | eqeltri 2697 | . . . . . . 7 ⊢ 𝑃 ∈ Word {𝑋, 𝑌} |
20 | wrdf 13310 | . . . . . . 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 6194 | . . . . . . . . 9 ⊢ (#‘𝑃) = (#‘〈“𝑋𝑌𝑋”〉) |
23 | s3len 13639 | . . . . . . . . 9 ⊢ (#‘〈“𝑋𝑌𝑋”〉) = 3 | |
24 | 22, 23 | eqtr2i 2645 | . . . . . . . 8 ⊢ 3 = (#‘𝑃) |
25 | 24 | oveq2i 6661 | . . . . . . 7 ⊢ (0..^3) = (0..^(#‘𝑃)) |
26 | 25 | feq2i 6037 | . . . . . 6 ⊢ (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌}) |
27 | 21, 26 | mpbir 221 | . . . . 5 ⊢ 𝑃:(0..^3)⟶{𝑋, 𝑌} |
28 | 12 | fveq2i 6194 | . . . . . . . . 9 ⊢ (#‘𝐹) = (#‘〈“00”〉) |
29 | s2len 13634 | . . . . . . . . 9 ⊢ (#‘〈“00”〉) = 2 | |
30 | 28, 29 | eqtri 2644 | . . . . . . . 8 ⊢ (#‘𝐹) = 2 |
31 | 30 | oveq2i 6661 | . . . . . . 7 ⊢ (0...(#‘𝐹)) = (0...2) |
32 | 3z 11410 | . . . . . . . . 9 ⊢ 3 ∈ ℤ | |
33 | fzoval 12471 | . . . . . . . . 9 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
34 | 32, 33 | ax-mp 5 | . . . . . . . 8 ⊢ (0..^3) = (0...(3 − 1)) |
35 | 3m1e2 11137 | . . . . . . . . 9 ⊢ (3 − 1) = 2 | |
36 | 35 | oveq2i 6661 | . . . . . . . 8 ⊢ (0...(3 − 1)) = (0...2) |
37 | 34, 36 | eqtr2i 2645 | . . . . . . 7 ⊢ (0...2) = (0..^3) |
38 | 31, 37 | eqtri 2644 | . . . . . 6 ⊢ (0...(#‘𝐹)) = (0..^3) |
39 | 38 | feq2i 6037 | . . . . 5 ⊢ (𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌}) |
40 | 27, 39 | mpbir 221 | . . . 4 ⊢ 𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} |
41 | 2, 12, 5, 6, 14 | wlk2v2elem2 27016 | . . . 4 ⊢ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} |
42 | 13, 40, 41 | 3pm3.2i 1239 | . . 3 ⊢ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
43 | 1 | fveq2i 6194 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) |
44 | prex 4909 | . . . . . 6 ⊢ {𝑋, 𝑌} ∈ V | |
45 | s1cli 13384 | . . . . . . 7 ⊢ 〈“{𝑋, 𝑌}”〉 ∈ Word V | |
46 | 2, 45 | eqeltri 2697 | . . . . . 6 ⊢ 𝐼 ∈ Word V |
47 | opvtxfv 25884 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌}) | |
48 | 44, 46, 47 | mp2an 708 | . . . . 5 ⊢ (Vtx‘〈{𝑋, 𝑌}, 𝐼〉) = {𝑋, 𝑌} |
49 | 43, 48 | eqtr2i 2645 | . . . 4 ⊢ {𝑋, 𝑌} = (Vtx‘𝐺) |
50 | 1 | fveq2i 6194 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) |
51 | opiedgfv 25887 | . . . . . 6 ⊢ (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼) | |
52 | 44, 46, 51 | mp2an 708 | . . . . 5 ⊢ (iEdg‘〈{𝑋, 𝑌}, 𝐼〉) = 𝐼 |
53 | 50, 52 | eqtr2i 2645 | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) |
54 | 49, 53 | upgriswlk 26537 | . . 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 248 | . 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 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 {cpr 4179 〈cop 4183 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 2c2 11070 3c3 11071 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 〈“cs1 13294 〈“cs2 13586 〈“cs3 13587 Vtxcvtx 25874 iEdgciedg 25875 UPGraph cupgr 25975 USPGraph cuspgr 26043 Walkscwlks 26492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-uspgr 26045 df-wlks 26495 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |