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Theorem wlknewwlksn 27021
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 26759 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 wlkn0 26751 . . . . . 6 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊) ≠ ∅)
31, 2sylbi 207 . . . . 5 (𝑊 ∈ (Walks‘𝐺) → (2nd𝑊) ≠ ∅)
43adantl 467 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ≠ ∅)
5 eqid 2771 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
6 eqid 2771 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
7 eqid 2771 . . . . . . 7 (1st𝑊) = (1st𝑊)
8 eqid 2771 . . . . . . 7 (2nd𝑊) = (2nd𝑊)
95, 6, 7, 8wlkelwrd 26763 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)))
10 ffz0iswrd 13528 . . . . . . 7 ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1110adantl 467 . . . . . 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 467 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
14 eqid 2771 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
1514upgrwlkvtxedg 26776 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^(♯‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16 wlklenvm1 26752 . . . . . . . . 9 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (♯‘(1st𝑊)) = ((♯‘(2nd𝑊)) − 1))
1716adantl 467 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (♯‘(1st𝑊)) = ((♯‘(2nd𝑊)) − 1))
1817oveq2d 6812 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (0..^(♯‘(1st𝑊))) = (0..^((♯‘(2nd𝑊)) − 1)))
1918raleqdv 3293 . . . . . 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 581 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ∀𝑖 ∈ (0..^((♯‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
224, 13, 213jca 1122 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2322adantr 466 . 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 468 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → 𝑁 ∈ ℕ0)
25 oveq2 6804 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) = 𝑁 → (0...(♯‘(1st𝑊))) = (0...𝑁))
2625adantl 467 . . . . . . . . . . . 12 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1st𝑊)) = 𝑁) → (0...(♯‘(1st𝑊))) = (0...𝑁))
2726feq2d 6170 . . . . . . . . . . 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 439 . . . . . . . . 9 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((♯‘(1st𝑊)) = 𝑁 → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3029adantld 478 . . . . . . . 8 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3130imp 393 . . . . . . 7 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺))
32 ffz0hash 13433 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3324, 31, 32syl2an2 666 . . . . . 6 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3433ex 397 . . . . 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 467 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → (♯‘(2nd𝑊)) = (𝑁 + 1)))
3736imp 393 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3824adantl 467 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → 𝑁 ∈ ℕ0)
39 iswwlksn 26966 . . . 4 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2nd𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2nd𝑊)) = (𝑁 + 1))))
405, 14iswwlks 26964 . . . . . 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 615 . . . 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 692 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 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  c0 4063  {cpr 4319   class class class wbr 4787  dom cdm 5250  wf 6026  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  0cc0 10142  1c1 10143   + caddc 10145  cmin 10472  0cn0 11499  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13487  Vtxcvtx 26095  iEdgciedg 26096  Edgcedg 26160  UPGraphcupgr 26196  Walkscwlks 26727  WWalkscwwlks 26953   WWalksN cwwlksn 26954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-n0 11500  df-xnn0 11571  df-z 11585  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-edg 26161  df-uhgr 26174  df-upgr 26198  df-wlks 26730  df-wwlks 26958  df-wwlksn 26959
This theorem is referenced by:  wlknwwlksnfunOLD  27022  wlkwwlkfunOLD  27030
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