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Theorem clwwlknwwlksn 29291
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 29286 . 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 12220 . . . . . . . . . . . . . 14 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
5 npcan1 11639 . . . . . . . . . . . . . 14 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
64, 5syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„• β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
76eqcomd 2739 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ 𝑁 = ((𝑁 βˆ’ 1) + 1))
87eqeq2d 2744 . . . . . . . . . . 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 1444 . . . . . . . . 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 1120 . . . . . . 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 482 . . . . 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 1348 . . . 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 29280 . . . . 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 2733 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
20 eqid 2733 . . . . . 6 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
2119, 20isclwwlk 29237 . . . . 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 625 . . . 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 12513 . . . 4 (𝑁 ∈ β„• β†’ (𝑁 βˆ’ 1) ∈ β„•0)
2519, 20iswwlksnx 29094 . . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆ…c0 4323  {cpr 4631  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•cn 12212  β„•0cn0 12472  ..^cfzo 13627  β™―chash 14290  Word cword 14464  lastSclsw 14512  Vtxcvtx 28256  Edgcedg 28307   WWalksN cwwlksn 29080  ClWWalkscclwwlk 29234   ClWWalksN cclwwlkn 29277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wwlks 29084  df-wwlksn 29085  df-clwwlk 29235  df-clwwlkn 29278
This theorem is referenced by:  clwwnrepclwwn  29597
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