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Theorem clwwlknun 29403
Description: The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
Hypothesis
Ref Expression
clwwlknun.v 𝑉 = (Vtxβ€˜πΊ)
Assertion
Ref Expression
clwwlknun (𝐺 ∈ USGraph β†’ (𝑁 ClWWalksN 𝐺) = βˆͺ π‘₯ ∈ 𝑉 (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁))
Distinct variable groups:   π‘₯,𝐺   π‘₯,𝑁   π‘₯,𝑉

Proof of Theorem clwwlknun
Dummy variables 𝑦 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 5001 . . 3 (𝑦 ∈ βˆͺ π‘₯ ∈ 𝑉 (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁) ↔ βˆƒπ‘₯ ∈ 𝑉 𝑦 ∈ (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁))
2 isclwwlknon 29382 . . . . 5 (𝑦 ∈ (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯))
32rexbii 3094 . . . 4 (βˆƒπ‘₯ ∈ 𝑉 𝑦 ∈ (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁) ↔ βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯))
4 simpl 483 . . . . . 6 ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯) β†’ 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
54rexlimivw 3151 . . . . 5 (βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯) β†’ 𝑦 ∈ (𝑁 ClWWalksN 𝐺))
6 clwwlknun.v . . . . . . . . 9 𝑉 = (Vtxβ€˜πΊ)
7 eqid 2732 . . . . . . . . 9 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
86, 7clwwlknp 29328 . . . . . . . 8 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) β†’ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ)))
98anim2i 617 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) β†’ (𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))))
107, 6usgrpredgv 28492 . . . . . . . . . . . . 13 ((𝐺 ∈ USGraph ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ)) β†’ ((lastSβ€˜π‘¦) ∈ 𝑉 ∧ (π‘¦β€˜0) ∈ 𝑉))
1110ex 413 . . . . . . . . . . . 12 (𝐺 ∈ USGraph β†’ ({(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((lastSβ€˜π‘¦) ∈ 𝑉 ∧ (π‘¦β€˜0) ∈ 𝑉)))
12 simpr 485 . . . . . . . . . . . 12 (((lastSβ€˜π‘¦) ∈ 𝑉 ∧ (π‘¦β€˜0) ∈ 𝑉) β†’ (π‘¦β€˜0) ∈ 𝑉)
1311, 12syl6com 37 . . . . . . . . . . 11 ({(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ) β†’ (𝐺 ∈ USGraph β†’ (π‘¦β€˜0) ∈ 𝑉))
14133ad2ant3 1135 . . . . . . . . . 10 (((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ)) β†’ (𝐺 ∈ USGraph β†’ (π‘¦β€˜0) ∈ 𝑉))
1514impcom 408 . . . . . . . . 9 ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) β†’ (π‘¦β€˜0) ∈ 𝑉)
16 simpr 485 . . . . . . . . . . . 12 (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) ∧ π‘₯ = (π‘¦β€˜0)) β†’ π‘₯ = (π‘¦β€˜0))
1716eqcomd 2738 . . . . . . . . . . 11 (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) ∧ π‘₯ = (π‘¦β€˜0)) β†’ (π‘¦β€˜0) = π‘₯)
1817biantrud 532 . . . . . . . . . 10 (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) ∧ π‘₯ = (π‘¦β€˜0)) β†’ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯)))
1918bicomd 222 . . . . . . . . 9 (((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) ∧ π‘₯ = (π‘¦β€˜0)) β†’ ((𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
2015, 19rspcedv 3605 . . . . . . . 8 ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) β†’ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) β†’ βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯)))
2120adantld 491 . . . . . . 7 ((𝐺 ∈ USGraph ∧ ((𝑦 ∈ Word 𝑉 ∧ (β™―β€˜π‘¦) = 𝑁) ∧ βˆ€π‘– ∈ (0..^(𝑁 βˆ’ 1)){(π‘¦β€˜π‘–), (π‘¦β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘¦), (π‘¦β€˜0)} ∈ (Edgβ€˜πΊ))) β†’ ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) β†’ βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯)))
229, 21mpcom 38 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)) β†’ βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯))
2322ex 413 . . . . 5 (𝐺 ∈ USGraph β†’ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) β†’ βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯)))
245, 23impbid2 225 . . . 4 (𝐺 ∈ USGraph β†’ (βˆƒπ‘₯ ∈ 𝑉 (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∧ (π‘¦β€˜0) = π‘₯) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
253, 24bitrid 282 . . 3 (𝐺 ∈ USGraph β†’ (βˆƒπ‘₯ ∈ 𝑉 𝑦 ∈ (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁) ↔ 𝑦 ∈ (𝑁 ClWWalksN 𝐺)))
261, 25bitr2id 283 . 2 (𝐺 ∈ USGraph β†’ (𝑦 ∈ (𝑁 ClWWalksN 𝐺) ↔ 𝑦 ∈ βˆͺ π‘₯ ∈ 𝑉 (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁)))
2726eqrdv 2730 1 (𝐺 ∈ USGraph β†’ (𝑁 ClWWalksN 𝐺) = βˆͺ π‘₯ ∈ 𝑉 (π‘₯(ClWWalksNOnβ€˜πΊ)𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  {cpr 4630  βˆͺ ciun 4997  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115   βˆ’ cmin 11446  ..^cfzo 13629  β™―chash 14292  Word cword 14466  lastSclsw 14514  Vtxcvtx 28294  Edgcedg 28345  USGraphcusgr 28447   ClWWalksN cclwwlkn 29315  ClWWalksNOncclwwlknon 29378
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-xnn0 12547  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-edg 28346  df-umgr 28381  df-usgr 28449  df-clwwlk 29273  df-clwwlkn 29316  df-clwwlknon 29379
This theorem is referenced by:  numclwwlk4  29677
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