Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1wlk2v2e Structured version   Visualization version   GIF version

Theorem 1wlk2v2e 41324
 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.)
Hypotheses
Ref Expression
1wlk2v2e.i 𝐼 = ⟨“{𝑋, 𝑌}”⟩
1wlk2v2e.f 𝐹 = ⟨“00”⟩
1wlk2v2e.x 𝑋 ∈ V
1wlk2v2e.y 𝑌 ∈ V
1wlk2v2e.p 𝑃 = ⟨“𝑋𝑌𝑋”⟩
1wlk2v2e.g 𝐺 = ⟨{𝑋, 𝑌}, 𝐼
Assertion
Ref Expression
1wlk2v2e 𝐹(1Walks‘𝐺)𝑃

Proof of Theorem 1wlk2v2e
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 1wlk2v2e.g . . . . 5 𝐺 = ⟨{𝑋, 𝑌}, 𝐼
2 1wlk2v2e.i . . . . . 6 𝐼 = ⟨“{𝑋, 𝑌}”⟩
32opeq2i 4344 . . . . 5 ⟨{𝑋, 𝑌}, 𝐼⟩ = ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩
41, 3eqtri 2632 . . . 4 𝐺 = ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩
5 1wlk2v2e.x . . . . 5 𝑋 ∈ V
6 1wlk2v2e.y . . . . 5 𝑌 ∈ V
7 uspgr2v1e2w 40477 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩ ∈ USPGraph )
85, 6, 7mp2an 704 . . . 4 ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩ ∈ USPGraph
94, 8eqeltri 2684 . . 3 𝐺 ∈ USPGraph
10 uspgrupgr 40406 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph )
119, 10ax-mp 5 . 2 𝐺 ∈ UPGraph
12 1wlk2v2e.f . . 3 𝐹 = ⟨“00”⟩
13 s2cli 13475 . . 3 ⟨“00”⟩ ∈ Word V
1412, 13eqeltri 2684 . 2 𝐹 ∈ Word V
15 1wlk2v2e.p . . 3 𝑃 = ⟨“𝑋𝑌𝑋”⟩
16 s3cli 13476 . . 3 ⟨“𝑋𝑌𝑋”⟩ ∈ Word V
1715, 16eqeltri 2684 . 2 𝑃 ∈ Word V
182, 121wlk2v2elem1 41322 . . . 4 𝐹 ∈ Word dom 𝐼
195elexi 3186 . . . . . . . . . 10 𝑋 ∈ V
2019prid1 4241 . . . . . . . . 9 𝑋 ∈ {𝑋, 𝑌}
216elexi 3186 . . . . . . . . . 10 𝑌 ∈ V
2221prid2 4242 . . . . . . . . 9 𝑌 ∈ {𝑋, 𝑌}
23 s3cl 13474 . . . . . . . . 9 ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → ⟨“𝑋𝑌𝑋”⟩ ∈ Word {𝑋, 𝑌})
2420, 22, 20, 23mp3an 1416 . . . . . . . 8 ⟨“𝑋𝑌𝑋”⟩ ∈ Word {𝑋, 𝑌}
2515, 24eqeltri 2684 . . . . . . 7 𝑃 ∈ Word {𝑋, 𝑌}
26 wrdf 13165 . . . . . . 7 (𝑃 ∈ Word {𝑋, 𝑌} → 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌})
2725, 26ax-mp 5 . . . . . 6 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌}
2815fveq2i 6106 . . . . . . . . 9 (#‘𝑃) = (#‘⟨“𝑋𝑌𝑋”⟩)
29 s3len 13489 . . . . . . . . 9 (#‘⟨“𝑋𝑌𝑋”⟩) = 3
3028, 29eqtr2i 2633 . . . . . . . 8 3 = (#‘𝑃)
3130oveq2i 6560 . . . . . . 7 (0..^3) = (0..^(#‘𝑃))
3231feq2i 5950 . . . . . 6 (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌})
3327, 32mpbir 220 . . . . 5 𝑃:(0..^3)⟶{𝑋, 𝑌}
3412fveq2i 6106 . . . . . . . . 9 (#‘𝐹) = (#‘⟨“00”⟩)
35 s2len 13484 . . . . . . . . 9 (#‘⟨“00”⟩) = 2
3634, 35eqtri 2632 . . . . . . . 8 (#‘𝐹) = 2
3736oveq2i 6560 . . . . . . 7 (0...(#‘𝐹)) = (0...2)
38 3z 11287 . . . . . . . . 9 3 ∈ ℤ
39 fzoval 12340 . . . . . . . . 9 (3 ∈ ℤ → (0..^3) = (0...(3 − 1)))
4038, 39ax-mp 5 . . . . . . . 8 (0..^3) = (0...(3 − 1))
41 3m1e2 11014 . . . . . . . . 9 (3 − 1) = 2
4241oveq2i 6560 . . . . . . . 8 (0...(3 − 1)) = (0...2)
4340, 42eqtr2i 2633 . . . . . . 7 (0...2) = (0..^3)
4437, 43eqtri 2632 . . . . . 6 (0...(#‘𝐹)) = (0..^3)
4544feq2i 5950 . . . . 5 (𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌})
4633, 45mpbir 220 . . . 4 𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌}
472, 12, 5, 6, 151wlk2v2elem2 41323 . . . 4 𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}
4818, 46, 473pm3.2i 1232 . . 3 (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
491fveq2i 6106 . . . . 5 (Vtx‘𝐺) = (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩)
50 prex 4836 . . . . . 6 {𝑋, 𝑌} ∈ V
51 s1cli 13237 . . . . . . 7 ⟨“{𝑋, 𝑌}”⟩ ∈ Word V
522, 51eqeltri 2684 . . . . . 6 𝐼 ∈ Word V
53 opvtxfv 25681 . . . . . 6 (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩) = {𝑋, 𝑌})
5450, 52, 53mp2an 704 . . . . 5 (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩) = {𝑋, 𝑌}
5549, 54eqtr2i 2633 . . . 4 {𝑋, 𝑌} = (Vtx‘𝐺)
561fveq2i 6106 . . . . 5 (iEdg‘𝐺) = (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩)
57 opiedgfv 25684 . . . . . 6 (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩) = 𝐼)
5850, 52, 57mp2an 704 . . . . 5 (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩) = 𝐼
5956, 58eqtr2i 2633 . . . 4 𝐼 = (iEdg‘𝐺)
6055, 59upgriswlk 40849 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
6148, 60mpbiri 247 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word V ∧ 𝑃 ∈ Word V) → 𝐹(1Walks‘𝐺)𝑃)
6211, 14, 17, 61mp3an 1416 1 𝐹(1Walks‘𝐺)𝑃
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {cpr 4127  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  3c3 10948  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  ⟨“cs1 13149  ⟨“cs2 13437  ⟨“cs3 13438  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747   USPGraph cuspgr 40378  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-rmo 2904  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-2o 7448  df-oadd 7451  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-cda 8873  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-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-vtx 25675  df-iedg 25676  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801 This theorem is referenced by: (None)
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