Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wlknewwlksn Structured version   Visualization version   GIF version

Theorem wlknewwlksn 41084
 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 ∈ (1Walks‘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 1wlkcpr 40833 . . . . . 6 (𝑊 ∈ (1Walks‘𝐺) ↔ (1st𝑊)(1Walks‘𝐺)(2nd𝑊))
2 1wlkn0 40825 . . . . . 6 ((1st𝑊)(1Walks‘𝐺)(2nd𝑊) → (2nd𝑊) ≠ ∅)
31, 2sylbi 206 . . . . 5 (𝑊 ∈ (1Walks‘𝐺) → (2nd𝑊) ≠ ∅)
43adantl 481 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → (2nd𝑊) ≠ ∅)
5 eqid 2610 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
6 eqid 2610 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
7 eqid 2610 . . . . . . 7 (1st𝑊) = (1st𝑊)
8 eqid 2610 . . . . . . 7 (2nd𝑊) = (2nd𝑊)
95, 6, 7, 81wlkelwrd 40837 . . . . . 6 (𝑊 ∈ (1Walks‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)))
10 ffz0iswrd 13187 . . . . . . 7 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1110adantl 481 . . . . . 6 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
129, 11syl 17 . . . . 5 (𝑊 ∈ (1Walks‘𝐺) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
1312adantl 481 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → (2nd𝑊) ∈ Word (Vtx‘𝐺))
14 eqid 2610 . . . . . . 7 (Edg‘𝐺) = (Edg‘𝐺)
1514upgr1wlkvtxedg 40853 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(1Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^(#‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
16 1wlklenvm1 40826 . . . . . . . . 9 ((1st𝑊)(1Walks‘𝐺)(2nd𝑊) → (#‘(1st𝑊)) = ((#‘(2nd𝑊)) − 1))
1716adantl 481 . . . . . . . 8 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(1Walks‘𝐺)(2nd𝑊)) → (#‘(1st𝑊)) = ((#‘(2nd𝑊)) − 1))
1817oveq2d 6565 . . . . . . 7 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(1Walks‘𝐺)(2nd𝑊)) → (0..^(#‘(1st𝑊))) = (0..^((#‘(2nd𝑊)) − 1)))
1918raleqdv 3121 . . . . . 6 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(1Walks‘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 221 . . . . 5 ((𝐺 ∈ UPGraph ∧ (1st𝑊)(1Walks‘𝐺)(2nd𝑊)) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
211, 20sylan2b 491 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
224, 13, 213jca 1235 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
2322adantr 480 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
24 simpl 472 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → 𝑁 ∈ ℕ0)
25 oveq2 6557 . . . . . . . . . . . . 13 ((#‘(1st𝑊)) = 𝑁 → (0...(#‘(1st𝑊))) = (0...𝑁))
2625adantl 481 . . . . . . . . . . . 12 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → (0...(#‘(1st𝑊))) = (0...𝑁))
2726feq2d 5944 . . . . . . . . . . 11 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ↔ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
2827biimpd 218 . . . . . . . . . 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 482 . . . . . . . 8 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)))
3130imp 444 . . . . . . 7 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺))
32 ffz0hash 13088 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (2nd𝑊):(0...𝑁)⟶(Vtx‘𝐺)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3324, 31, 32syl2an2 871 . . . . . 6 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3433ex 449 . . . . 5 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
359, 34syl 17 . . . 4 (𝑊 ∈ (1Walks‘𝐺) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
3635adantl 481 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁) → (#‘(2nd𝑊)) = (𝑁 + 1)))
3736imp 444 . 2 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (#‘(2nd𝑊)) = (𝑁 + 1))
3824adantl 481 . . 3 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → 𝑁 ∈ ℕ0)
39 iswwlksn 41041 . . . 4 (𝑁 ∈ ℕ0 → ((2nd𝑊) ∈ (𝑁 WWalkSN 𝐺) ↔ ((2nd𝑊) ∈ (WWalkS‘𝐺) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
405, 14iswwlks 41039 . . . . . 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 737 . . . 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 267 . . 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 ∈ (1Walks‘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 959 1 (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalkSN 𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  {cpr 4127   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792  1Walksc1wlks 40796  WWalkScwwlks 41028   WWalkSN cwwlksn 41029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  wlknwwlksnfun  41085  wlkwwlkfun  41092
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