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Theorem clwlkfclwwlk1hash 26369
 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.)
Hypotheses
Ref Expression
clwlkfclwwlk.1 𝐴 = (1st𝑐)
clwlkfclwwlk.2 𝐵 = (2nd𝑐)
clwlkfclwwlk.c 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlkfclwwlk1hash (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
Distinct variable groups:   𝐸,𝑐   𝑁,𝑐   𝑉,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝐶(𝑐)   𝐹(𝑐)

Proof of Theorem clwlkfclwwlk1hash
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwlkfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
21rabeq2i 3170 . 2 (𝑐𝐶 ↔ (𝑐 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝐴) = 𝑁))
3 clwlkfclwwlk.1 . . . . 5 𝐴 = (1st𝑐)
4 clwlkfclwwlk.2 . . . . 5 𝐵 = (2nd𝑐)
53, 4clwlkcompim 26292 . . . 4 (𝑐 ∈ (𝑉 ClWalks 𝐸) → ((𝐴 ∈ Word dom 𝐸𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))))
6 lencl 13179 . . . . . 6 (𝐴 ∈ Word dom 𝐸 → (#‘𝐴) ∈ ℕ0)
7 ffn 5958 . . . . . 6 (𝐵:(0...(#‘𝐴))⟶𝑉𝐵 Fn (0...(#‘𝐴)))
8 fnfz0hash 13087 . . . . . . 7 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → (#‘𝐵) = ((#‘𝐴) + 1))
9 id 22 . . . . . . . . . 10 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℕ0)
10 peano2nn0 11210 . . . . . . . . . 10 ((#‘𝐴) ∈ ℕ0 → ((#‘𝐴) + 1) ∈ ℕ0)
11 nn0re 11178 . . . . . . . . . . 11 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ ℝ)
1211lep1d 10834 . . . . . . . . . 10 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ≤ ((#‘𝐴) + 1))
13 elfz2nn0 12300 . . . . . . . . . 10 ((#‘𝐴) ∈ (0...((#‘𝐴) + 1)) ↔ ((#‘𝐴) ∈ ℕ0 ∧ ((#‘𝐴) + 1) ∈ ℕ0 ∧ (#‘𝐴) ≤ ((#‘𝐴) + 1)))
149, 10, 12, 13syl3anbrc 1239 . . . . . . . . 9 ((#‘𝐴) ∈ ℕ0 → (#‘𝐴) ∈ (0...((#‘𝐴) + 1)))
15 oveq2 6557 . . . . . . . . . 10 ((#‘𝐵) = ((#‘𝐴) + 1) → (0...(#‘𝐵)) = (0...((#‘𝐴) + 1)))
1615eleq2d 2673 . . . . . . . . 9 ((#‘𝐵) = ((#‘𝐴) + 1) → ((#‘𝐴) ∈ (0...(#‘𝐵)) ↔ (#‘𝐴) ∈ (0...((#‘𝐴) + 1))))
1714, 16syl5ibrcom 236 . . . . . . . 8 ((#‘𝐴) ∈ ℕ0 → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵))))
1817adantr 480 . . . . . . 7 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → ((#‘𝐵) = ((#‘𝐴) + 1) → (#‘𝐴) ∈ (0...(#‘𝐵))))
198, 18mpd 15 . . . . . 6 (((#‘𝐴) ∈ ℕ0𝐵 Fn (0...(#‘𝐴))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
206, 7, 19syl2an 493 . . . . 5 ((𝐴 ∈ Word dom 𝐸𝐵:(0...(#‘𝐴))⟶𝑉) → (#‘𝐴) ∈ (0...(#‘𝐵)))
2120adantr 480 . . . 4 (((𝐴 ∈ Word dom 𝐸𝐵:(0...(#‘𝐴))⟶𝑉) ∧ (∀𝑖 ∈ (0..^(#‘𝐴))(𝐸‘(𝐴𝑖)) = {(𝐵𝑖), (𝐵‘(𝑖 + 1))} ∧ (𝐵‘0) = (𝐵‘(#‘𝐴)))) → (#‘𝐴) ∈ (0...(#‘𝐵)))
225, 21syl 17 . . 3 (𝑐 ∈ (𝑉 ClWalks 𝐸) → (#‘𝐴) ∈ (0...(#‘𝐵)))
2322adantr 480 . 2 ((𝑐 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘𝐴) = 𝑁) → (#‘𝐴) ∈ (0...(#‘𝐵)))
242, 23sylbi 206 1 (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150   ClWalks cclwlk 26275 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-clwlk 26278 This theorem is referenced by:  clwlkfclwwlk2sswd  26370  clwlkfclwwlk  26371
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