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Theorem wlknewwlksn 26676
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 ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalksN 𝐺))

Proof of Theorem wlknewwlksn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkcpr 26428 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 wlkn0 26420 . . . . . 6 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊) ≠ ∅)
31, 2sylbi 207 . . . . 5 (𝑊 ∈ (Walks‘𝐺) → (2nd𝑊) ≠ ∅)
43adantl 482 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ≠ ∅)
5 eqid 2621 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
6 eqid 2621 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
7 eqid 2621 . . . . . . 7 (1st𝑊) = (1st𝑊)
8 eqid 2621 . . . . . . 7 (2nd𝑊) = (2nd𝑊)
95, 6, 7, 8wlkelwrd 26432 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)))
10 ffz0iswrd 13287 . . . . . . 7 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1110adantl 482 . . . . . 6 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
129, 11syl 17 . . . . 5 (𝑊 ∈ (Walks‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1312adantl 482 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
14 eqid 2621 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
1514upgrwlkvtxedg 26444 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^(#‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16 wlklenvm1 26421 . . . . . . . . 9 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (#‘(1st𝑊)) = ((#‘(2nd𝑊)) − 1))
1716adantl 482 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (#‘(1st𝑊)) = ((#‘(2nd𝑊)) − 1))
1817oveq2d 6631 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (0..^(#‘(1st𝑊))) = (0..^((#‘(2nd𝑊)) − 1)))
1918raleqdv 3137 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (∀𝑖 ∈ (0..^(#‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2015, 19mpbid 222 . . . . 5 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
211, 20sylan2b 492 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
224, 13, 213jca 1240 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2322adantr 481 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
24 simpl 473 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → 𝑁 ∈ ℕ0)
25 oveq2 6623 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) = 𝑁 → (0...(#‘(1st𝑊))) = (0...𝑁))
2625adantl 482 . . . . . . . . . . . 12 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → (0...(#‘(1st𝑊))) = (0...𝑁))
2726feq2d 5998 . . . . . . . . . . 11 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
2827biimpd 219 . . . . . . . . . 10 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
2928impancom 456 . . . . . . . . 9 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((#‘(1st𝑊)) = 𝑁 → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3029adantld 483 . . . . . . . 8 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3130imp 445 . . . . . . 7 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺))
32 ffz0hash 13185 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3324, 31, 32syl2an2 874 . . . . . 6 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3433ex 450 . . . . 5 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
359, 34syl 17 . . . 4 (𝑊 ∈ (Walks‘𝐺) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
3635adantl 482 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
3736imp 445 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3824adantl 482 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → 𝑁 ∈ ℕ0)
39 iswwlksn 26633 . . . 4 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2nd𝑊) ∈ (WWalks‘𝐺) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
405, 14iswwlks 26631 . . . . . 6 ((2nd𝑊) ∈ (WWalks‘𝐺) ↔ ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4140a1i 11 . . . . 5 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (WWalks‘𝐺) ↔ ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))))
4241anbi1d 740 . . . 4 (𝑁 ∈ ℕ0 → (((2nd𝑊) ∈ (WWalks‘𝐺) ∧ (#‘(2nd𝑊)) = (𝑁 + 1)) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4339, 42bitrd 268 . . 3 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4438, 43syl 17 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4523, 37, 44mpbir2and 956 1 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalksN 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  c0 3897  {cpr 4157   class class class wbr 4623  dom cdm 5084  wf 5853  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  0cc0 9896  1c1 9897   + caddc 9899  cmin 10226  0cn0 11252  ...cfz 12284  ..^cfzo 12422  #chash 13073  Word cword 13246  Vtxcvtx 25808  iEdgciedg 25809  Edgcedg 25873   UPGraph cupgr 25905  Walkscwlks 26396  WWalkscwwlks 26620   WWalksN cwwlksn 26621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-hash 13074  df-word 13254  df-edg 25874  df-uhgr 25883  df-upgr 25907  df-wlks 26399  df-wwlks 26625  df-wwlksn 26626
This theorem is referenced by:  wlknwwlksnfun  26677  wlkwwlkfun  26684
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