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Theorem clwwlknwwlksn 29559
Description: A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 βˆ’ 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then βŸ¨β€œπ‘Žπ‘π‘β€βŸ© ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if π‘Ž β‰  𝑐) of length 2, and βŸ¨β€œπ‘Žπ‘π‘π‘Žβ€βŸ© ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) (Revised by AV, 22-Mar-2022.)
Assertion
Ref Expression
clwwlknwwlksn (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 βˆ’ 1) WWalksN 𝐺))
Proof of Theorem clwwlknwwlksn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwwlknnn 29554 . 2 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ 𝑁 ∈ β„•)
2 idd 24 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ)))
3 idd 24 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
4 nncn 12225 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
5 npcan1 11644 . . . . . . . . . . . . . 14 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
64, 5syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
76eqcomd 2737 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ 𝑁 = ((𝑁 βˆ’ 1) + 1))
87eqeq2d 2742 . . . . . . . . . . 11 (𝑁 ∈ β„• β†’ ((β™―β€˜π‘Š) = 𝑁 ↔ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1)))
98biimpd 228 . . . . . . . . . 10 (𝑁 ∈ β„• β†’ ((β™―β€˜π‘Š) = 𝑁 β†’ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1)))
102, 3, 93anim123d 1442 . . . . . . . . 9 (𝑁 ∈ β„• β†’ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
1110com12 32 . . . . . . . 8 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
12113exp 1118 . . . . . . 7 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = 𝑁 β†’ (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))))
1312a1dd 50 . . . . . 6 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = 𝑁 β†’ (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1)))))))
1413adantr 480 . . . . 5 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) β†’ ({(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ) β†’ ((β™―β€˜π‘Š) = 𝑁 β†’ (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1)))))))
15143imp1 1346 . . . 4 ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (𝑁 ∈ β„• β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
1615com12 32 . . 3 (𝑁 ∈ β„• β†’ ((((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 𝑁) β†’ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
17 isclwwlkn 29548 . . . . 5 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ↔ (π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁))
1817a1i 11 . . . 4 (𝑁 ∈ β„• β†’ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ↔ (π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁)))
19 eqid 2731 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
20 eqid 2731 . . . . . 6 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
2119, 20isclwwlk 29505 . . . . 5 (π‘Š ∈ (ClWWalksβ€˜πΊ) ↔ ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)))
2221anbi1i 623 . . . 4 ((π‘Š ∈ (ClWWalksβ€˜πΊ) ∧ (β™―β€˜π‘Š) = 𝑁) ↔ (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 𝑁))
2318, 22bitrdi 287 . . 3 (𝑁 ∈ β„• β†’ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) ↔ (((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ {(lastSβ€˜π‘Š), (π‘Šβ€˜0)} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜π‘Š) = 𝑁)))
24 nnm1nn0 12518 . . . 4 (𝑁 ∈ β„• β†’ (𝑁 βˆ’ 1) ∈ β„•0)
2519, 20iswwlksnx 29362 . . . 4 ((𝑁 βˆ’ 1) ∈ β„•0 β†’ (π‘Š ∈ ((𝑁 βˆ’ 1) WWalksN 𝐺) ↔ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
2624, 25syl 17 . . 3 (𝑁 ∈ β„• β†’ (π‘Š ∈ ((𝑁 βˆ’ 1) WWalksN 𝐺) ↔ (π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜π‘Š) βˆ’ 1)){(π‘Šβ€˜π‘–), (π‘Šβ€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ∧ (β™―β€˜π‘Š) = ((𝑁 βˆ’ 1) + 1))))
2716, 23, 263imtr4d 294 . 2 (𝑁 ∈ β„• β†’ (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 βˆ’ 1) WWalksN 𝐺)))
281, 27mpcom 38 1 (π‘Š ∈ (𝑁 ClWWalksN 𝐺) β†’ π‘Š ∈ ((𝑁 βˆ’ 1) WWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆ…c0 4322  {cpr 4630  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11112  0cc0 11114  1c1 11115   + caddc 11117   βˆ’ cmin 11449  β„•cn 12217  β„•0cn0 12477  ..^cfzo 13632  β™―chash 14295  Word cword 14469  lastSclsw 14517  Vtxcvtx 28524  Edgcedg 28575   WWalksN cwwlksn 29348  ClWWalkscclwwlk 29502   ClWWalksN cclwwlkn 29545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-oadd 8474  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-wwlks 29352  df-wwlksn 29353  df-clwwlk 29503  df-clwwlkn 29546
This theorem is referenced by:  clwwnrepclwwn  29865
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