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Theorem reuccats1 13332
 Description: A set of words having the length of a given word increased by 1 contains a unique word with the given word as prefix if there is a unique symbol which extends the given word to be a word of the set. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
Assertion
Ref Expression
reuccats1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
Distinct variable groups:   𝑣,𝑉,𝑥   𝑣,𝑊,𝑥   𝑣,𝑋,𝑥

Proof of Theorem reuccats1
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2676 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
2 fveq2 6103 . . . . 5 (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
32eqeq1d 2612 . . . 4 (𝑥 = 𝑦 → ((#‘𝑥) = ((#‘𝑊) + 1) ↔ (#‘𝑦) = ((#‘𝑊) + 1)))
41, 3anbi12d 743 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))))
54cbvralv 3147 . 2 (∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1)))
6 s1eq 13233 . . . . . 6 (𝑣 = 𝑢 → ⟨“𝑣”⟩ = ⟨“𝑢”⟩)
76oveq2d 6565 . . . . 5 (𝑣 = 𝑢 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
87eleq1d 2672 . . . 4 (𝑣 = 𝑢 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
98reu8 3369 . . 3 (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)))
10 simprl 790 . . . . . 6 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
11 simp-4l 802 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → 𝑊 ∈ Word 𝑉)
12 simpr 476 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → 𝑥𝑋)
1310adantr 480 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
14 simplrr 797 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))
15 simp-4r 803 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1)))
16 reuccats1lem 13331 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑥𝑋 ∧ (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋) ∧ (∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢) ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1)))) → (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
1711, 12, 13, 14, 15, 16syl32anc 1326 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
18 oveq1 6556 . . . . . . . . . . 11 (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 substr ⟨0, (#‘𝑊)⟩) = ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (#‘𝑊)⟩))
19 simpl 472 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
20 s1cl 13235 . . . . . . . . . . . . . . 15 (𝑣𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
2119, 20anim12i 588 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
2221adantr 480 . . . . . . . . . . . . 13 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
2322adantr 480 . . . . . . . . . . . 12 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
24 swrdccat1 13309 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
2523, 24syl 17 . . . . . . . . . . 11 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (#‘𝑊)⟩) = 𝑊)
2618, 25sylan9eqr 2666 . . . . . . . . . 10 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → (𝑥 substr ⟨0, (#‘𝑊)⟩) = 𝑊)
2726eqcomd 2616 . . . . . . . . 9 ((((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) ∧ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)) → 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))
2827ex 449 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
2917, 28impbid 201 . . . . . . 7 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
3029ralrimiva 2949 . . . . . 6 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∀𝑥𝑋 (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
31 reu6i 3364 . . . . . 6 (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥𝑋 (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))
3210, 30, 31syl2anc 691 . . . . 5 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))
3332ex 449 . . . 4 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
3433rexlimdva 3013 . . 3 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) → (∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
359, 34syl5bi 231 . 2 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
365, 35sylan2b 491 1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149   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-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  df-concat 13156  df-s1 13157  df-substr 13158 This theorem is referenced by:  numclwlk2lem2f1o  26632  av-numclwlk2lem2f1o  41535
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