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Theorem wwlksnextfun 26861
Description: Lemma for wwlksnextbij 26865. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextbij0.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
wwlksnextbij.r 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij.f 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
Assertion
Ref Expression
wwlksnextfun (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑊   𝑡,𝐷   𝑛,𝐸   𝑤,𝐸   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉   𝑤,𝑉   𝑛,𝑊   𝑡,𝑛
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐸(𝑡)   𝐹(𝑤,𝑡,𝑛)   𝐺(𝑡,𝑛)   𝑁(𝑛)   𝑉(𝑡)   𝑊(𝑡)

Proof of Theorem wwlksnextfun
StepHypRef Expression
1 fveq2 6229 . . . . . . 7 (𝑤 = 𝑡 → (#‘𝑤) = (#‘𝑡))
21eqeq1d 2653 . . . . . 6 (𝑤 = 𝑡 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑡) = (𝑁 + 2)))
3 oveq1 6697 . . . . . . 7 (𝑤 = 𝑡 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑡 substr ⟨0, (𝑁 + 1)⟩))
43eqeq1d 2653 . . . . . 6 (𝑤 = 𝑡 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
5 fveq2 6229 . . . . . . . 8 (𝑤 = 𝑡 → ( lastS ‘𝑤) = ( lastS ‘𝑡))
65preq2d 4307 . . . . . . 7 (𝑤 = 𝑡 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
76eleq1d 2715 . . . . . 6 (𝑤 = 𝑡 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
82, 4, 73anbi123d 1439 . . . . 5 (𝑤 = 𝑡 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸) ↔ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)))
9 wwlksnextbij0.d . . . . 5 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
108, 9elrab2 3399 . . . 4 (𝑡𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)))
11 simpll 805 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
12 nn0re 11339 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
13 2re 11128 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
1413a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
15 nn0ge0 11356 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
16 2pos 11150 . . . . . . . . . . . . . . . . 17 0 < 2
1716a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 < 2)
1812, 14, 15, 17addgegt0d 10639 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1918ad2antlr 763 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
20 breq2 4689 . . . . . . . . . . . . . . 15 ((#‘𝑡) = (𝑁 + 2) → (0 < (#‘𝑡) ↔ 0 < (𝑁 + 2)))
2120adantl 481 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → (0 < (#‘𝑡) ↔ 0 < (𝑁 + 2)))
2219, 21mpbird 247 . . . . . . . . . . . . 13 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (#‘𝑡))
23 hashgt0n0 13194 . . . . . . . . . . . . 13 ((𝑡 ∈ Word 𝑉 ∧ 0 < (#‘𝑡)) → 𝑡 ≠ ∅)
2411, 22, 23syl2anc 694 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
2511, 24jca 553 . . . . . . . . . . 11 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
2625expcom 450 . . . . . . . . . 10 ((#‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
27263ad2ant1 1102 . . . . . . . . 9 (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2827expd 451 . . . . . . . 8 (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸) → (𝑡 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))))
2928impcom 445 . . . . . . 7 ((𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
3029impcom 445 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
31 lswcl 13388 . . . . . 6 ((𝑡 ∈ Word 𝑉𝑡 ≠ ∅) → ( lastS ‘𝑡) ∈ 𝑉)
3230, 31syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → ( lastS ‘𝑡) ∈ 𝑉)
33 simprr3 1131 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)
3432, 33jca 553 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
3510, 34sylan2b 491 . . 3 ((𝑁 ∈ ℕ0𝑡𝐷) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
36 preq2 4301 . . . . 5 (𝑛 = ( lastS ‘𝑡) → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
3736eleq1d 2715 . . . 4 (𝑛 = ( lastS ‘𝑡) → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
38 wwlksnextbij.r . . . 4 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}
3937, 38elrab2 3399 . . 3 (( lastS ‘𝑡) ∈ 𝑅 ↔ (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸))
4035, 39sylibr 224 . 2 ((𝑁 ∈ ℕ0𝑡𝐷) → ( lastS ‘𝑡) ∈ 𝑅)
41 wwlksnextbij.f . 2 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
4240, 41fmptd 6425 1 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  c0 3948  {cpr 4212  cop 4216   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  2c2 11108  0cn0 11330  #chash 13157  Word cword 13323   lastS clsw 13324   substr csubstr 13327  Vtxcvtx 25919  Edgcedg 25984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-lsw 13332
This theorem is referenced by:  wwlksnextinj  26862  wwlksnextsur  26863
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