MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wlknewwlksn Structured version   Visualization version   GIF version

Theorem wlknewwlksn 27825
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 27570 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))
2 wlkn0 27562 . . . . . 6 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (2nd𝑊) ≠ ∅)
31, 2sylbi 220 . . . . 5 (𝑊 ∈ (Walks‘𝐺) → (2nd𝑊) ≠ ∅)
43adantl 485 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ≠ ∅)
5 eqid 2738 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
6 eqid 2738 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
7 eqid 2738 . . . . . . 7 (1st𝑊) = (1st𝑊)
8 eqid 2738 . . . . . . 7 (2nd𝑊) = (2nd𝑊)
95, 6, 7, 8wlkelwrd 27574 . . . . . 6 (𝑊 ∈ (Walks‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)))
10 ffz0iswrd 13982 . . . . . . 7 ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1110adantl 485 . . . . . 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 485 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
14 eqid 2738 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
1514upgrwlkvtxedg 27586 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^(♯‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16 wlklenvm1 27563 . . . . . . . . 9 ((1st𝑊)(Walks‘𝐺)(2nd𝑊) → (♯‘(1st𝑊)) = ((♯‘(2nd𝑊)) − 1))
1716adantl 485 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (♯‘(1st𝑊)) = ((♯‘(2nd𝑊)) − 1))
1817oveq2d 7186 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → (0..^(♯‘(1st𝑊))) = (0..^((♯‘(2nd𝑊)) − 1)))
1918raleqdv 3316 . . . . . 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 235 . . . . 5 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^((♯‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
211, 20sylan2b 597 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ∀𝑖 ∈ (0..^((♯‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
224, 13, 213jca 1129 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((♯‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2322adantr 484 . 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 486 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → 𝑁 ∈ ℕ0)
25 oveq2 7178 . . . . . . . . . . . . 13 ((♯‘(1st𝑊)) = 𝑁 → (0...(♯‘(1st𝑊))) = (0...𝑁))
2625adantl 485 . . . . . . . . . . . 12 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1st𝑊)) = 𝑁) → (0...(♯‘(1st𝑊))) = (0...𝑁))
2726feq2d 6490 . . . . . . . . . . 11 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
2827biimpd 232 . . . . . . . . . 10 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (♯‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
2928impancom 455 . . . . . . . . 9 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((♯‘(1st𝑊)) = 𝑁 → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3029adantld 494 . . . . . . . 8 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3130imp 410 . . . . . . 7 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺))
32 ffz0hash 13897 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3324, 31, 32syl2an2 686 . . . . . 6 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3433ex 416 . . . . 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 485 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁) → (♯‘(2nd𝑊)) = (𝑁 + 1)))
3736imp 410 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (♯‘(2nd𝑊)) = (𝑁 + 1))
3824adantl 485 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → 𝑁 ∈ ℕ0)
39 iswwlksn 27776 . . . 4 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (𝑁 WWalksN 𝐺) ↔ ((2nd𝑊) ∈ (WWalks‘𝐺) ∧ (♯‘(2nd𝑊)) = (𝑁 + 1))))
405, 14iswwlks 27774 . . . . . 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 633 . . . 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 282 . . 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 713 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wne 2934  wral 3053  c0 4211  {cpr 4518   class class class wbr 5030  dom cdm 5525  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  0cc0 10615  1c1 10616   + caddc 10618  cmin 10948  0cn0 11976  ...cfz 12981  ..^cfzo 13124  chash 13782  Word cword 13955  Vtxcvtx 26941  iEdgciedg 26942  Edgcedg 26992  UPGraphcupgr 27025  Walkscwlks 27538  WWalkscwwlks 27763   WWalksN cwwlksn 27764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-oadd 8135  df-er 8320  df-map 8439  df-pm 8440  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-dju 9403  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-n0 11977  df-xnn0 12049  df-z 12063  df-uz 12325  df-fz 12982  df-fzo 13125  df-hash 13783  df-word 13956  df-edg 26993  df-uhgr 27003  df-upgr 27027  df-wlks 27541  df-wwlks 27768  df-wwlksn 27769
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator