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Theorem wlknewwlksn 29141
Description: If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)
Assertion
Ref Expression
wlknewwlksn (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ (2nd β€˜π‘Š) ∈ (𝑁 WWalksN 𝐺))

Proof of Theorem wlknewwlksn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkcpr 28886 . . . . . 6 (π‘Š ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š))
2 wlkn0 28878 . . . . . 6 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ (2nd β€˜π‘Š) β‰  βˆ…)
31, 2sylbi 216 . . . . 5 (π‘Š ∈ (Walksβ€˜πΊ) β†’ (2nd β€˜π‘Š) β‰  βˆ…)
43adantl 483 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ (2nd β€˜π‘Š) β‰  βˆ…)
5 eqid 2733 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
6 eqid 2733 . . . . . . 7 (iEdgβ€˜πΊ) = (iEdgβ€˜πΊ)
7 eqid 2733 . . . . . . 7 (1st β€˜π‘Š) = (1st β€˜π‘Š)
8 eqid 2733 . . . . . . 7 (2nd β€˜π‘Š) = (2nd β€˜π‘Š)
95, 6, 7, 8wlkelwrd 28890 . . . . . 6 (π‘Š ∈ (Walksβ€˜πΊ) β†’ ((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)))
10 ffz0iswrd 14491 . . . . . . 7 ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) β†’ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ))
1110adantl 483 . . . . . 6 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) β†’ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ))
129, 11syl 17 . . . . 5 (π‘Š ∈ (Walksβ€˜πΊ) β†’ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ))
1312adantl 483 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ))
14 eqid 2733 . . . . . . 7 (Edgβ€˜πΊ) = (Edgβ€˜πΊ)
1514upgrwlkvtxedg 28902 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š)) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜(1st β€˜π‘Š))){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
16 wlklenvm1 28879 . . . . . . . . 9 ((1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š) β†’ (β™―β€˜(1st β€˜π‘Š)) = ((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1))
1716adantl 483 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š)) β†’ (β™―β€˜(1st β€˜π‘Š)) = ((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1))
1817oveq2d 7425 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š)) β†’ (0..^(β™―β€˜(1st β€˜π‘Š))) = (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)))
1918raleqdv 3326 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜(1st β€˜π‘Š))){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ) ↔ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2015, 19mpbid 231 . . . . 5 ((𝐺 ∈ UPGraph ∧ (1st β€˜π‘Š)(Walksβ€˜πΊ)(2nd β€˜π‘Š)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
211, 20sylan2b 595 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))
224, 13, 213jca 1129 . . 3 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ ((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
2322adantr 482 . 2 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ ((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
24 simpl 484 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ 𝑁 ∈ β„•0)
25 oveq2 7417 . . . . . . . . . . . . 13 ((β™―β€˜(1st β€˜π‘Š)) = 𝑁 β†’ (0...(β™―β€˜(1st β€˜π‘Š))) = (0...𝑁))
2625adantl 483 . . . . . . . . . . . 12 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ (0...(β™―β€˜(1st β€˜π‘Š))) = (0...𝑁))
2726feq2d 6704 . . . . . . . . . . 11 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) ↔ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ)))
2827biimpd 228 . . . . . . . . . 10 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ ((2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ) β†’ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ)))
2928impancom 453 . . . . . . . . 9 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) β†’ ((β™―β€˜(1st β€˜π‘Š)) = 𝑁 β†’ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ)))
3029adantld 492 . . . . . . . 8 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) β†’ ((𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ)))
3130imp 408 . . . . . . 7 ((((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ))
32 ffz0hash 14406 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ (2nd β€˜π‘Š):(0...𝑁)⟢(Vtxβ€˜πΊ)) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))
3324, 31, 32syl2an2 685 . . . . . 6 ((((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))
3433ex 414 . . . . 5 (((1st β€˜π‘Š) ∈ Word dom (iEdgβ€˜πΊ) ∧ (2nd β€˜π‘Š):(0...(β™―β€˜(1st β€˜π‘Š)))⟢(Vtxβ€˜πΊ)) β†’ ((𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1)))
359, 34syl 17 . . . 4 (π‘Š ∈ (Walksβ€˜πΊ) β†’ ((𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1)))
3635adantl 483 . . 3 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ ((𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1)))
3736imp 408 . 2 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))
3824adantl 483 . . 3 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ 𝑁 ∈ β„•0)
39 iswwlksn 29092 . . . 4 (𝑁 ∈ β„•0 β†’ ((2nd β€˜π‘Š) ∈ (𝑁 WWalksN 𝐺) ↔ ((2nd β€˜π‘Š) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))))
405, 14iswwlks 29090 . . . . . 6 ((2nd β€˜π‘Š) ∈ (WWalksβ€˜πΊ) ↔ ((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)))
4140a1i 11 . . . . 5 (𝑁 ∈ β„•0 β†’ ((2nd β€˜π‘Š) ∈ (WWalksβ€˜πΊ) ↔ ((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ))))
4241anbi1d 631 . . . 4 (𝑁 ∈ β„•0 β†’ (((2nd β€˜π‘Š) ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1)) ↔ (((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))))
4339, 42bitrd 279 . . 3 (𝑁 ∈ β„•0 β†’ ((2nd β€˜π‘Š) ∈ (𝑁 WWalksN 𝐺) ↔ (((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))))
4438, 43syl 17 . 2 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ ((2nd β€˜π‘Š) ∈ (𝑁 WWalksN 𝐺) ↔ (((2nd β€˜π‘Š) β‰  βˆ… ∧ (2nd β€˜π‘Š) ∈ Word (Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^((β™―β€˜(2nd β€˜π‘Š)) βˆ’ 1)){((2nd β€˜π‘Š)β€˜π‘–), ((2nd β€˜π‘Š)β€˜(𝑖 + 1))} ∈ (Edgβ€˜πΊ)) ∧ (β™―β€˜(2nd β€˜π‘Š)) = (𝑁 + 1))))
4523, 37, 44mpbir2and 712 1 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) ∧ (𝑁 ∈ β„•0 ∧ (β™―β€˜(1st β€˜π‘Š)) = 𝑁)) β†’ (2nd β€˜π‘Š) ∈ (𝑁 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   class class class wbr 5149  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  0cc0 11110  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•0cn0 12472  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307  UPGraphcupgr 28340  Walkscwlks 28853  WWalkscwwlks 29079   WWalksN cwwlksn 29080
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-ifp 1063  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-2o 8467  df-oadd 8470  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  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-2 12275  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-edg 28308  df-uhgr 28318  df-upgr 28342  df-wlks 28856  df-wwlks 29084  df-wwlksn 29085
This theorem is referenced by: (None)
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