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Theorem reuccats1 13413
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 2692 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
2 fveq2 6150 . . . . 5 (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
32eqeq1d 2628 . . . 4 (𝑥 = 𝑦 → ((#‘𝑥) = ((#‘𝑊) + 1) ↔ (#‘𝑦) = ((#‘𝑊) + 1)))
41, 3anbi12d 746 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))))
54cbvralv 3164 . 2 (∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (#‘𝑥) = ((#‘𝑊) + 1)) ↔ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1)))
6 s1eq 13314 . . . . . 6 (𝑣 = 𝑢 → ⟨“𝑣”⟩ = ⟨“𝑢”⟩)
76oveq2d 6621 . . . . 5 (𝑣 = 𝑢 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
87eleq1d 2688 . . . 4 (𝑣 = 𝑢 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
98reu8 3389 . . 3 (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)))
10 simprl 793 . . . . . 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 805 . . . . . . . . 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 477 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → 𝑥𝑋)
1310adantr 481 . . . . . . . . 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 800 . . . . . . . . 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 806 . . . . . . . . 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 13412 . . . . . . . . 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 1331 . . . . . . . 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 6612 . . . . . . . . . . 11 (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 substr ⟨0, (#‘𝑊)⟩) = ((𝑊 ++ ⟨“𝑣”⟩) substr ⟨0, (#‘𝑊)⟩))
19 simpl 473 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
20 s1cl 13316 . . . . . . . . . . . . . . 15 (𝑣𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
2119, 20anim12i 589 . . . . . . . . . . . . . 14 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) ∧ 𝑣𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
2221adantr 481 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 13390 . . . . . . . . . . . 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 2682 . . . . . . . . . 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 2632 . . . . . . . . 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 450 . . . . . . . 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 202 . . . . . . 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 2965 . . . . . 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 3384 . . . . . 6 (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥𝑋 (𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩))
3210, 30, 31syl2anc 692 . . . . 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 450 . . . 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 3029 . . 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 232 . 2 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (#‘𝑦) = ((#‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 substr ⟨0, (#‘𝑊)⟩)))
365, 35sylan2b 492 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  ∃!wreu 2914  cop 4159  cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  #chash 13054  Word cword 13225   ++ cconcat 13227  ⟨“cs1 13228   substr csubstr 13229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-xnn0 11309  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-hash 13055  df-word 13233  df-lsw 13234  df-concat 13235  df-s1 13236  df-substr 13237
This theorem is referenced by:  numclwlk2lem2f1o  27087
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