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Theorem clwlksfclwwlk1hash 26826
 Description: The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlksfclwwlk1hash (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝐶(𝑐)   𝐹(𝑐)

Proof of Theorem clwlksfclwwlk1hash
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
21rabeq2i 3183 . 2 (𝑐𝐶 ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁))
3 eqid 2621 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
4 eqid 2621 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
5 clwlksfclwwlk.1 . . . . 5 𝐴 = (1st𝑐)
6 clwlksfclwwlk.2 . . . . 5 𝐵 = (2nd𝑐)
73, 4, 5, 6clwlkcompim 26545 . . . 4 (𝑐 ∈ (ClWalks‘𝐺) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))if-((𝐵𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖)}, {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴𝑖))) ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))))
8 lencl 13263 . . . . . 6 (𝐴 ∈ Word dom (iEdg‘𝐺) → (#‘𝐴) ∈ ℕ0)
9 ffn 6002 . . . . . 6 (𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺) → 𝐵 Fn (0...(#‘𝐴)))
10 fnfz0hash 13168 . . . . . . 7 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → (#‘𝐵) = ((#‘𝐴) + 1))
11 nn0fz0 12378 . . . . . . . . . 10 ((#‘𝐴) ∈ ℕ0 ↔ (#‘𝐴) ∈ (0...(#‘𝐴)))
12 fzelp1 12335 . . . . . . . . . 10 ((#‘𝐴) ∈ (0...(#‘𝐴)) → (#‘𝐴) ∈ (0...((#‘𝐴) + 1)))
1311, 12sylbi 207 . . . . . . . . 9 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...((#‘𝐴) + 1)))
14 oveq2 6612 . . . . . . . . . 10 ((#‘𝐵) = ((#‘𝐴) + 1) → (0...(#‘𝐵)) = (0...((#‘𝐴) + 1)))
1514eleq2d 2684 . . . . . . . . 9 ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ (#‘𝐴) ∈ (0...((#‘𝐴) + 1))))
1613, 15syl5ibrcom 237 . . . . . . . 8 ((#‘𝐴) ∈ ℕ0 → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵))))
1716adantr 481 . . . . . . 7 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵))))
1810, 17mpd 15 . . . . . 6 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
198, 9, 18syl2an 494 . . . . 5 ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) → (#‘𝐴) ∈ (0...(#‘𝐵)))
2019adantr 481 . . . 4 (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(#‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))if-((𝐵𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴𝑖)) = {(𝐵𝑖)}, {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴𝑖))) ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
217, 20syl 17 . . 3 (𝑐 ∈ (ClWalks‘𝐺) → (#‘𝐴) ∈ (0...(#‘𝐵)))
2221adantr 481 . 2 ((𝑐 ∈ (ClWalks‘𝐺) ∧ (#‘𝐴) = 𝑁) → (#‘𝐴) ∈ (0...(#‘𝐵)))
232, 22sylbi 207 1 (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  if-wif 1011   = wceq 1480   ∈ wcel 1987  ∀wral 2907  {crab 2911   ⊆ wss 3555  {csn 4148  {cpr 4150  ⟨cop 4154   ↦ cmpt 4673  dom cdm 5074   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  0cc0 9880  1c1 9881   + caddc 9883  ℕ0cn0 11236  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230   substr csubstr 13234  Vtxcvtx 25774  iEdgciedg 25775  ClWalkscclwlks 26535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-wlks 26365  df-clwlks 26536 This theorem is referenced by:  clwlksfclwwlk2sswd  26827  clwlksfclwwlk  26828
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