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Theorem vfwlkniswwlkn 26234
 Description: If the edge function of a walk has length n, then the vertex function of the walk is a word representing the walk as word of length n. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
Assertion
Ref Expression
vfwlkniswwlkn ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))

Proof of Theorem vfwlkniswwlkn
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkcpr 26057 . . . . 5 (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊))
2 wlkn0 26055 . . . . 5 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → (2nd𝑊) ≠ ∅)
31, 2sylbi 206 . . . 4 (𝑊 ∈ (𝑉 Walks 𝐸) → (2nd𝑊) ≠ ∅)
4 wlkelwrd 26058 . . . . 5 (𝑊 ∈ (𝑉 Walks 𝐸) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
5 ffz0iswrd 13187 . . . . . 6 ((2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 → (2nd𝑊) ∈ Word 𝑉)
65adantl 481 . . . . 5 (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) → (2nd𝑊) ∈ Word 𝑉)
74, 6syl 17 . . . 4 (𝑊 ∈ (𝑉 Walks 𝐸) → (2nd𝑊) ∈ Word 𝑉)
8 edgwlk 26059 . . . . . 6 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → ∀𝑖 ∈ (0..^(#‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸)
9 wlklenvm1 26060 . . . . . . . 8 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → (#‘(1st𝑊)) = ((#‘(2nd𝑊)) − 1))
109oveq2d 6565 . . . . . . 7 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → (0..^(#‘(1st𝑊))) = (0..^((#‘(2nd𝑊)) − 1)))
1110raleqdv 3121 . . . . . 6 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → (∀𝑖 ∈ (0..^(#‘(1st𝑊))){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸))
128, 11mpbid 221 . . . . 5 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸)
131, 12sylbi 206 . . . 4 (𝑊 ∈ (𝑉 Walks 𝐸) → ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸)
143, 7, 133jca 1235 . . 3 (𝑊 ∈ (𝑉 Walks 𝐸) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸))
1514ad2antrl 760 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸))
16 id 22 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
17 oveq2 6557 . . . . . . . . . . 11 ((#‘(1st𝑊)) = 𝑁 → (0...(#‘(1st𝑊))) = (0...𝑁))
1817adantl 481 . . . . . . . . . 10 (((1st𝑊) ∈ Word dom 𝐸 ∧ (#‘(1st𝑊)) = 𝑁) → (0...(#‘(1st𝑊))) = (0...𝑁))
1918feq2d 5944 . . . . . . . . 9 (((1st𝑊) ∈ Word dom 𝐸 ∧ (#‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 ↔ (2nd𝑊):(0...𝑁)⟶𝑉))
2019biimpd 218 . . . . . . . 8 (((1st𝑊) ∈ Word dom 𝐸 ∧ (#‘(1st𝑊)) = 𝑁) → ((2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 → (2nd𝑊):(0...𝑁)⟶𝑉))
2120impancom 455 . . . . . . 7 (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) → ((#‘(1st𝑊)) = 𝑁 → (2nd𝑊):(0...𝑁)⟶𝑉))
2221imp 444 . . . . . 6 ((((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) ∧ (#‘(1st𝑊)) = 𝑁) → (2nd𝑊):(0...𝑁)⟶𝑉)
23 ffz0hash 13088 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (2nd𝑊):(0...𝑁)⟶𝑉) → (#‘(2nd𝑊)) = (𝑁 + 1))
2416, 22, 23syl2anr 494 . . . . 5 (((((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) ∧ (#‘(1st𝑊)) = 𝑁) ∧ 𝑁 ∈ ℕ0) → (#‘(2nd𝑊)) = (𝑁 + 1))
2524ex 449 . . . 4 ((((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉) ∧ (#‘(1st𝑊)) = 𝑁) → (𝑁 ∈ ℕ0 → (#‘(2nd𝑊)) = (𝑁 + 1)))
264, 25sylan 487 . . 3 ((𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁) → (𝑁 ∈ ℕ0 → (#‘(2nd𝑊)) = (𝑁 + 1)))
2726impcom 445 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (#‘(2nd𝑊)) = (𝑁 + 1))
28 wlkbprop 26051 . . . . . . . 8 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → ((#‘(1st𝑊)) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ ((1st𝑊) ∈ V ∧ (2nd𝑊) ∈ V)))
2928simp2d 1067 . . . . . . 7 ((1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
301, 29sylbi 206 . . . . . 6 (𝑊 ∈ (𝑉 Walks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
31 simpll 786 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → 𝑉 ∈ V)
32 simpr 476 . . . . . . . . 9 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
3332adantr 480 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → 𝐸 ∈ V)
34 simpr 476 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
3531, 33, 343jca 1235 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
3635ex 449 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ0 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0)))
3730, 36syl 17 . . . . 5 (𝑊 ∈ (𝑉 Walks 𝐸) → (𝑁 ∈ ℕ0 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0)))
3837adantr 480 . . . 4 ((𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁) → (𝑁 ∈ ℕ0 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0)))
3938impcom 445 . . 3 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
40 iswwlkn 26212 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ ((2nd𝑊) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
41 iswwlk 26211 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((2nd𝑊) ∈ (𝑉 WWalks 𝐸) ↔ ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸)))
42413adant3 1074 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((2nd𝑊) ∈ (𝑉 WWalks 𝐸) ↔ ((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸)))
4342anbi1d 737 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (((2nd𝑊) ∈ (𝑉 WWalks 𝐸) ∧ (#‘(2nd𝑊)) = (𝑁 + 1)) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4440, 43bitrd 267 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4539, 44syl 17 . 2 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (((2nd𝑊) ≠ ∅ ∧ (2nd𝑊) ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘(2nd𝑊)) − 1)){((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ∈ ran 𝐸) ∧ (#‘(2nd𝑊)) = (𝑁 + 1))))
4615, 27, 45mpbir2and 959 1 ((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶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   Walks cwalk 26026   WWalks cwwlk 26205   WWalksN cwwlkn 26206 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-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-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-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-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wlknwwlknfun  26238  wlkiswwlkfun  26245
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