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Theorem wwlksnextfun 27170
Description: Lemma for wwlksnextbij 27179. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
Hypotheses
Ref Expression
wwlksnextbij0.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij0.e 𝐸 = (Edg‘𝐺)
wwlksnextbij0.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
wwlksnextbij0.r 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
wwlksnextbij0.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 fveqeq2 6420 . . . . . 6 (𝑤 = 𝑡 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑡) = (𝑁 + 2)))
2 oveq1 6885 . . . . . . 7 (𝑤 = 𝑡 → (𝑤 prefix (𝑁 + 1)) = (𝑡 prefix (𝑁 + 1)))
32eqeq1d 2801 . . . . . 6 (𝑤 = 𝑡 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑡 prefix (𝑁 + 1)) = 𝑊))
4 fveq2 6411 . . . . . . . 8 (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
54preq2d 4464 . . . . . . 7 (𝑤 = 𝑡 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑡)})
65eleq1d 2863 . . . . . 6 (𝑤 = 𝑡 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
71, 3, 63anbi123d 1561 . . . . 5 (𝑤 = 𝑡 → (((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸) ↔ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
8 wwlksnextbij0.d . . . . 5 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}
97, 8elrab2 3560 . . . 4 (𝑡𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
10 simpll 784 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
11 nn0re 11590 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
12 2re 11387 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
1312a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
14 nn0ge0 11607 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
15 2pos 11423 . . . . . . . . . . . . . . . . 17 0 < 2
1615a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 < 2)
1711, 13, 14, 16addgegt0d 10893 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1817ad2antlr 719 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
19 breq2 4847 . . . . . . . . . . . . . . 15 ((♯‘𝑡) = (𝑁 + 2) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
2019adantl 474 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
2118, 20mpbird 249 . . . . . . . . . . . . 13 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (♯‘𝑡))
22 hashgt0n0 13406 . . . . . . . . . . . . 13 ((𝑡 ∈ Word 𝑉 ∧ 0 < (♯‘𝑡)) → 𝑡 ≠ ∅)
2310, 21, 22syl2anc 580 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
2410, 23jca 508 . . . . . . . . . . 11 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
2524expcom 403 . . . . . . . . . 10 ((♯‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
26253ad2ant1 1164 . . . . . . . . 9 (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2726expd 405 . . . . . . . 8 (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → (𝑡 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))))
2827impcom 397 . . . . . . 7 ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2928impcom 397 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
30 lswcl 13588 . . . . . 6 ((𝑡 ∈ Word 𝑉𝑡 ≠ ∅) → (lastS‘𝑡) ∈ 𝑉)
3129, 30syl 17 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → (lastS‘𝑡) ∈ 𝑉)
32 simprr3 1292 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)
3331, 32jca 508 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
349, 33sylan2b 588 . . 3 ((𝑁 ∈ ℕ0𝑡𝐷) → ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
35 preq2 4458 . . . . 5 (𝑛 = (lastS‘𝑡) → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), (lastS‘𝑡)})
3635eleq1d 2863 . . . 4 (𝑛 = (lastS‘𝑡) → ({(lastS‘𝑊), 𝑛} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
37 wwlksnextbij0.r . . . 4 𝑅 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}
3836, 37elrab2 3560 . . 3 ((lastS‘𝑡) ∈ 𝑅 ↔ ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
3934, 38sylibr 226 . 2 ((𝑁 ∈ ℕ0𝑡𝐷) → (lastS‘𝑡) ∈ 𝑅)
40 wwlksnextbij0.f . 2 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
4139, 40fmptd 6610 1 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  {crab 3093  c0 4115  {cpr 4370   class class class wbr 4843  cmpt 4922  wf 6097  cfv 6101  (class class class)co 6878  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   < clt 10363  2c2 11368  0cn0 11580  chash 13370  Word cword 13534  lastSclsw 13582   prefix cpfx 13713  Vtxcvtx 26231  Edgcedg 26282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-2 11376  df-n0 11581  df-xnn0 11653  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-lsw 13583
This theorem is referenced by:  wwlksnextinj  27171  wwlksnextsurj  27172
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