MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reuccatpfxs1 Structured version   Visualization version   GIF version

Theorem reuccatpfxs1 14312
Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)
Hypothesis
Ref Expression
reuccatpfxs1.1 𝑣𝑋
Assertion
Ref Expression
reuccatpfxs1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
Distinct variable groups:   𝑣,𝑉,𝑥   𝑣,𝑊,𝑥   𝑥,𝑋
Allowed substitution hint:   𝑋(𝑣)

Proof of Theorem reuccatpfxs1
Dummy variables 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1w 2820 . . . 4 (𝑥 = 𝑦 → (𝑥 ∈ Word 𝑉𝑦 ∈ Word 𝑉))
2 fveqeq2 6726 . . . 4 (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑊) + 1) ↔ (♯‘𝑦) = ((♯‘𝑊) + 1)))
31, 2anbi12d 634 . . 3 (𝑥 = 𝑦 → ((𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))))
43cbvralvw 3358 . 2 (∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1)) ↔ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
5 reuccatpfxs1.1 . . . . 5 𝑣𝑋
65nfel2 2922 . . . 4 𝑣(𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋
75nfel2 2922 . . . 4 𝑣(𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋
8 s1eq 14157 . . . . . 6 (𝑣 = 𝑥 → ⟨“𝑣”⟩ = ⟨“𝑥”⟩)
98oveq2d 7229 . . . . 5 (𝑣 = 𝑥 → (𝑊 ++ ⟨“𝑣”⟩) = (𝑊 ++ ⟨“𝑥”⟩))
109eleq1d 2822 . . . 4 (𝑣 = 𝑥 → ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋))
11 s1eq 14157 . . . . . 6 (𝑥 = 𝑢 → ⟨“𝑥”⟩ = ⟨“𝑢”⟩)
1211oveq2d 7229 . . . . 5 (𝑥 = 𝑢 → (𝑊 ++ ⟨“𝑥”⟩) = (𝑊 ++ ⟨“𝑢”⟩))
1312eleq1d 2822 . . . 4 (𝑥 = 𝑢 → ((𝑊 ++ ⟨“𝑥”⟩) ∈ 𝑋 ↔ (𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋))
146, 7, 10, 13reu8nf 3789 . . 3 (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ↔ ∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)))
15 nfv 1922 . . . . 5 𝑣 𝑊 ∈ Word 𝑉
16 nfv 1922 . . . . . 6 𝑣(𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
175, 16nfralw 3147 . . . . 5 𝑣𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))
1815, 17nfan 1907 . . . 4 𝑣(𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1)))
19 nfv 1922 . . . . 5 𝑣 𝑊 = (𝑥 prefix (♯‘𝑊))
205, 19nfreuw 3286 . . . 4 𝑣∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))
21 simprl 771 . . . . . 6 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋)
22 simpl 486 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → 𝑊 ∈ Word 𝑉)
2322ad2antrr 726 . . . . . . . . . 10 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → 𝑊 ∈ Word 𝑉)
2423anim1i 618 . . . . . . . . 9 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 ∈ Word 𝑉𝑥𝑋))
25 simplrr 778 . . . . . . . . 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))
26 simp-4r 784 . . . . . . . . 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)))
27 reuccatpfxs1lem 14311 . . . . . . . . 9 (((𝑊 ∈ Word 𝑉𝑥𝑋) ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢) ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (𝑊 = (𝑥 prefix (♯‘𝑊)) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
2824, 25, 26, 27syl3anc 1373 . . . . . . . 8 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 = (𝑥 prefix (♯‘𝑊)) → 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
29 oveq1 7220 . . . . . . . . . . 11 (𝑥 = (𝑊 ++ ⟨“𝑣”⟩) → (𝑥 prefix (♯‘𝑊)) = ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)))
30 s1cl 14159 . . . . . . . . . . . . . 14 (𝑣𝑉 → ⟨“𝑣”⟩ ∈ Word 𝑉)
3122, 30anim12i 616 . . . . . . . . . . . . 13 (((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) → (𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉))
3231ad2antrr 726 . . . . . . . . . . . 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))
33 pfxccat1 14267 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑉 ∧ ⟨“𝑣”⟩ ∈ Word 𝑉) → ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)) = 𝑊)
3432, 33syl 17 . . . . . . . . . . 11 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → ((𝑊 ++ ⟨“𝑣”⟩) prefix (♯‘𝑊)) = 𝑊)
3529, 34sylan9eqr 2800 . . . . . . . . . 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 prefix (♯‘𝑊)) = 𝑊)
3635eqcomd 2743 . . . . . . . . 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 prefix (♯‘𝑊)))
3736ex 416 . . . . . . . 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 prefix (♯‘𝑊))))
3828, 37impbid 215 . . . . . . 7 (((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) ∧ 𝑥𝑋) → (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
3938ralrimiva 3105 . . . . . 6 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∀𝑥𝑋 (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩)))
40 reu6i 3641 . . . . . 6 (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑥𝑋 (𝑊 = (𝑥 prefix (♯‘𝑊)) ↔ 𝑥 = (𝑊 ++ ⟨“𝑣”⟩))) → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))
4121, 39, 40syl2anc 587 . . . . 5 ((((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) ∧ 𝑣𝑉) ∧ ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢))) → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))
4241exp31 423 . . . 4 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (𝑣𝑉 → (((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊)))))
4318, 20, 42rexlimd 3236 . . 3 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃𝑣𝑉 ((𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 ∧ ∀𝑢𝑉 ((𝑊 ++ ⟨“𝑢”⟩) ∈ 𝑋𝑣 = 𝑢)) → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
4414, 43syl5bi 245 . 2 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑦𝑋 (𝑦 ∈ Word 𝑉 ∧ (♯‘𝑦) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
454, 44sylan2b 597 1 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑥𝑋 (𝑥 ∈ Word 𝑉 ∧ (♯‘𝑥) = ((♯‘𝑊) + 1))) → (∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ 𝑋 → ∃!𝑥𝑋 𝑊 = (𝑥 prefix (♯‘𝑊))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wnfc 2884  wral 3061  wrex 3062  ∃!wreu 3063  cfv 6380  (class class class)co 7213  1c1 10730   + caddc 10732  chash 13896  Word cword 14069   ++ cconcat 14125  ⟨“cs1 14152   prefix cpfx 14235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236
This theorem is referenced by:  reuccatpfxs1v  14313  numclwlk2lem2f1o  28462
  Copyright terms: Public domain W3C validator