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Theorem wwlkextfun 26257
 Description: Lemma 1 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
Hypotheses
Ref Expression
wwlkextbij.d 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
wwlkextbij.r 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}
wwlkextbij.f 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
Assertion
Ref Expression
wwlkextfun (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
Distinct variable groups:   𝑡,𝐷   𝑛,𝐸,𝑤   𝑡,𝑁,𝑤   𝑡,𝑅   𝑛,𝑉,𝑡,𝑤   𝑛,𝑊,𝑡,𝑤
Allowed substitution hints:   𝐷(𝑤,𝑛)   𝑅(𝑤,𝑛)   𝐸(𝑡)   𝐹(𝑤,𝑡,𝑛)   𝑁(𝑛)

Proof of Theorem wwlkextfun
StepHypRef Expression
1 fveq2 6103 . . . . . . 7 (𝑤 = 𝑡 → (#‘𝑤) = (#‘𝑡))
21eqeq1d 2612 . . . . . 6 (𝑤 = 𝑡 → ((#‘𝑤) = (𝑁 + 2) ↔ (#‘𝑡) = (𝑁 + 2)))
3 oveq1 6556 . . . . . . 7 (𝑤 = 𝑡 → (𝑤 substr ⟨0, (𝑁 + 1)⟩) = (𝑡 substr ⟨0, (𝑁 + 1)⟩))
43eqeq1d 2612 . . . . . 6 (𝑤 = 𝑡 → ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
5 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑡 → ( lastS ‘𝑤) = ( lastS ‘𝑡))
65preq2d 4219 . . . . . . 7 (𝑤 = 𝑡 → {( lastS ‘𝑊), ( lastS ‘𝑤)} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
76eleq1d 2672 . . . . . 6 (𝑤 = 𝑡 → ({( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))
82, 4, 73anbi123d 1391 . . . . 5 (𝑤 = 𝑡 → (((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸) ↔ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)))
9 wwlkextbij.d . . . . 5 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}
108, 9elrab2 3333 . . . 4 (𝑡𝐷 ↔ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)))
11 simpll 786 . . . . . . . . . . . 12 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 𝑡 ∈ Word 𝑉)
12 nn0re 11178 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0𝑁 ∈ ℝ)
13 2re 10967 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
1413a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 2 ∈ ℝ)
15 nn0ge0 11195 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 ≤ 𝑁)
16 2pos 10989 . . . . . . . . . . . . . . . . 17 0 < 2
1716a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → 0 < 2)
1812, 14, 15, 17addgegt0d 10480 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → 0 < (𝑁 + 2))
1918ad2antlr 759 . . . . . . . . . . . . . 14 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (𝑁 + 2))
20 breq2 4587 . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . 13 (((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) ∧ (#‘𝑡) = (𝑁 + 2)) → 0 < (#‘𝑡))
23 hashgt0n0 13017 . . . . . . . . . . . . 13 ((𝑡 ∈ Word 𝑉 ∧ 0 < (#‘𝑡)) → 𝑡 ≠ ∅)
2411, 22, 23syl2anc 691 . . . . . . . . . . . 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 1075 . . . . . . . . 9 (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸) → ((𝑡 ∈ Word 𝑉𝑁 ∈ ℕ0) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅)))
2827expd 451 . . . . . . . 8 (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 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)} ∈ ran 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)} ∈ ran 𝐸))) → (𝑡 ∈ Word 𝑉𝑡 ≠ ∅))
31 lswcl 13208 . . . . . 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)} ∈ ran 𝐸))) → ( lastS ‘𝑡) ∈ 𝑉)
33 simprr3 1104 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))) → {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸)
3432, 33jca 553 . . . 4 ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ Word 𝑉 ∧ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))
3510, 34sylan2b 491 . . 3 ((𝑁 ∈ ℕ0𝑡𝐷) → (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))
36 preq2 4213 . . . . 5 (𝑛 = ( lastS ‘𝑡) → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), ( lastS ‘𝑡)})
3736eleq1d 2672 . . . 4 (𝑛 = ( lastS ‘𝑡) → ({( lastS ‘𝑊), 𝑛} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))
38 wwlkextbij.r . . . 4 𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}
3937, 38elrab2 3333 . . 3 (( lastS ‘𝑡) ∈ 𝑅 ↔ (( lastS ‘𝑡) ∈ 𝑉 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ ran 𝐸))
4035, 39sylibr 223 . 2 ((𝑁 ∈ ℕ0𝑡𝐷) → ( lastS ‘𝑡) ∈ 𝑅)
41 wwlkextbij.f . 2 𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))
4240, 41fmptd 6292 1 (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  ∅c0 3874  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  2c2 10947  ℕ0cn0 11169  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150 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-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-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155 This theorem is referenced by:  wwlkextinj  26258  wwlkextsur  26259
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