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Theorem wwlksnextfun 27590
Description: Lemma for wwlksnextbij 27594. (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 6676 . . . . . 6 (𝑤 = 𝑡 → ((♯‘𝑤) = (𝑁 + 2) ↔ (♯‘𝑡) = (𝑁 + 2)))
2 oveq1 7155 . . . . . . 7 (𝑤 = 𝑡 → (𝑤 prefix (𝑁 + 1)) = (𝑡 prefix (𝑁 + 1)))
32eqeq1d 2828 . . . . . 6 (𝑤 = 𝑡 → ((𝑤 prefix (𝑁 + 1)) = 𝑊 ↔ (𝑡 prefix (𝑁 + 1)) = 𝑊))
4 fveq2 6667 . . . . . . . 8 (𝑤 = 𝑡 → (lastS‘𝑤) = (lastS‘𝑡))
54preq2d 4675 . . . . . . 7 (𝑤 = 𝑡 → {(lastS‘𝑊), (lastS‘𝑤)} = {(lastS‘𝑊), (lastS‘𝑡)})
65eleq1d 2902 . . . . . 6 (𝑤 = 𝑡 → ({(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸 ↔ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
71, 3, 63anbi123d 1429 . . . . 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 3687 . . . 4 (𝑡𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)))
10 simpll 763 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
11 nn0re 11895 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
12 2re 11700 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
1312a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
14 nn0ge0 11911 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
15 2pos 11729 . . . . . . . . . . . . . . . . 17 0 < 2
1615a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 < 2)
1711, 13, 14, 16addgegt0d 11202 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1817ad2antlr 723 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
19 breq2 5067 . . . . . . . . . . . . . . 15 ((♯‘𝑡) = (𝑁 + 2) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
2019adantl 482 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (0 < (♯‘𝑡) ↔ 0 < (𝑁 + 2)))
2118, 20mpbird 258 . . . . . . . . . . . . 13 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 0 < (♯‘𝑡))
22 hashgt0n0 13716 . . . . . . . . . . . . 13 ((𝑡 ∈ Word 𝑉 ∧ 0 < (♯‘𝑡)) → 𝑡 ≠ ∅)
2310, 21, 22syl2anc 584 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → 𝑡 ≠ ∅)
2410, 23jca 512 . . . . . . . . . . 11 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (♯‘𝑡) = (𝑁 + 2)) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
2524expcom 414 . . . . . . . . . 10 ((♯‘𝑡) = (𝑁 + 2) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
26253ad2ant1 1127 . . . . . . . . 9 (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2726expd 416 . . . . . . . 8 (((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸) → (𝑡 ∈ Word 𝑉 → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))))
2827impcom 408 . . . . . . 7 ((𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸)) → (𝑁 ∈ ℕ0 → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2928impcom 408 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((♯‘𝑡) = (𝑁 + 2) ∧ (𝑡 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
30 lswcl 13910 . . . . . 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 1217 . . . . 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 512 . . . 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 593 . . 3 ((𝑁 ∈ ℕ0𝑡𝐷) → ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
35 preq2 4669 . . . . 5 (𝑛 = (lastS‘𝑡) → {(lastS‘𝑊), 𝑛} = {(lastS‘𝑊), (lastS‘𝑡)})
3635eleq1d 2902 . . . 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 3687 . . 3 ((lastS‘𝑡) ∈ 𝑅 ↔ ((lastS‘𝑡) ∈ 𝑉 ∧ {(lastS‘𝑊), (lastS‘𝑡)} ∈ 𝐸))
3934, 38sylibr 235 . 2 ((𝑁 ∈ ℕ0𝑡𝐷) → (lastS‘𝑡) ∈ 𝑅)
40 wwlksnextbij0.f . 2 𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))
4139, 40fmptd 6874 1 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  {crab 3147  c0 4295  {cpr 4566   class class class wbr 5063  cmpt 5143  wf 6348  cfv 6352  (class class class)co 7148  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  2c2 11681  0cn0 11886  chash 13680  Word cword 13851  lastSclsw 13904   prefix cpfx 14022  Vtxcvtx 26695  Edgcedg 26746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-lsw 13905
This theorem is referenced by:  wwlksnextinj  27591  wwlksnextsurj  27592
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