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Theorem wwlknimp 26215
 Description: Implications for a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
Assertion
Ref Expression
wwlknimp (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
Distinct variable groups:   𝑖,𝐸   𝑖,𝑁   𝑖,𝑉   𝑖,𝑊

Proof of Theorem wwlknimp
StepHypRef Expression
1 wwlknprop 26214 . 2 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)))
2 iswwlkn 26212 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))))
3 simprr 792 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))) → (#‘𝑊) = (𝑁 + 1))
4 iswwlk 26211 . . . . . . . . . . . . . . . . 17 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ (𝑉 WWalks 𝐸) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
5 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
6 nn0cn 11179 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0𝑁 ∈ ℂ)
7 1cnd 9935 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ ℕ0 → 1 ∈ ℂ)
86, 7pncand 10272 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁)
98adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((𝑁 + 1) − 1) = 𝑁)
105, 9sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) − 1) = 𝑁)
1110oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑊) = (𝑁 + 1)) → (0..^((#‘𝑊) − 1)) = (0..^𝑁))
1211raleqdv 3121 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑊) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
1312biimpd 218 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑊) = (𝑁 + 1)) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
1413ex 449 . . . . . . . . . . . . . . . . . . . 20 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((#‘𝑊) = (𝑁 + 1) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
1514com23 84 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
1615impancom 455 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
17163adant1 1072 . . . . . . . . . . . . . . . . 17 ((𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) → (𝑁 ∈ ℕ0 → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
184, 17syl6bi 242 . . . . . . . . . . . . . . . 16 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ (𝑉 WWalks 𝐸) → (𝑁 ∈ ℕ0 → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))))
1918com23 84 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑉 WWalks 𝐸) → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))))
20193impia 1253 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ (𝑉 WWalks 𝐸) → ((#‘𝑊) = (𝑁 + 1) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
2120imp32 448 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))) → ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)
223, 21jca 553 . . . . . . . . . . . 12 (((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ∧ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
2322ex 449 . . . . . . . . . . 11 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
242, 23sylbid 229 . . . . . . . . . 10 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
25243expa 1257 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
2625ancoms 468 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
2726imp 444 . . . . . . 7 (((𝑁 ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)) → ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
2827anim2i 591 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ ((𝑁 ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
29 3anass 1035 . . . . . 6 ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ((#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
3028, 29sylibr 223 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ ((𝑁 ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) ∧ 𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁))) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
3130exp44 639 . . . 4 (𝑊 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))))
3231impcom 445 . . 3 ((𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))))
3332impcom 445 . 2 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))
341, 33mpcom 37 1 (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  {cpr 4127  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   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-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-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  wwlknimpb  26232  wwlknext  26252  wwlknextbi  26253  wwlknredwwlkn  26254  wwlknredwwlkn0  26255  wwlkextwrd  26256  wwlkextsur  26259  wwlkextproplem1  26269  wwlkextproplem2  26270  wwlkextproplem3  26271  clwwlkf1  26324  clwwlkvbij  26329  wwlkext2clwwlk  26331  rusgranumwlks  26483  numclwwlk2lem1  26629
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