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Theorem ccats1swrdeqrex 13411
Description: There exists a symbol such that its concatenation with the subword obtained by deleting the last symbol of a nonempty word results in the word itself. (Contributed by AV, 5-Oct-2018.) (Proof shortened by AV, 24-Oct-2018.)
Assertion
Ref Expression
ccats1swrdeqrex ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))
Distinct variable groups:   𝑈,𝑠   𝑉,𝑠   𝑊,𝑠

Proof of Theorem ccats1swrdeqrex
StepHypRef Expression
1 simp2 1060 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑈 ∈ Word 𝑉)
2 lencl 13258 . . . . . . . . . . 11 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
3 nn0p1gt0 11267 . . . . . . . . . . 11 ((#‘𝑊) ∈ ℕ0 → 0 < ((#‘𝑊) + 1))
42, 3syl 17 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → 0 < ((#‘𝑊) + 1))
543ad2ant1 1080 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 0 < ((#‘𝑊) + 1))
6 breq2 4622 . . . . . . . . . 10 ((#‘𝑈) = ((#‘𝑊) + 1) → (0 < (#‘𝑈) ↔ 0 < ((#‘𝑊) + 1)))
763ad2ant3 1082 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (0 < (#‘𝑈) ↔ 0 < ((#‘𝑊) + 1)))
85, 7mpbird 247 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 0 < (#‘𝑈))
9 hashneq0 13092 . . . . . . . . 9 (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) ↔ 𝑈 ≠ ∅))
1093ad2ant2 1081 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (0 < (#‘𝑈) ↔ 𝑈 ≠ ∅))
118, 10mpbid 222 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → 𝑈 ≠ ∅)
121, 11jca 554 . . . . . 6 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑈 ∈ Word 𝑉𝑈 ≠ ∅))
1312adantr 481 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)) → (𝑈 ∈ Word 𝑉𝑈 ≠ ∅))
14 lswcl 13289 . . . . 5 ((𝑈 ∈ Word 𝑉𝑈 ≠ ∅) → ( lastS ‘𝑈) ∈ 𝑉)
1513, 14syl 17 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)) → ( lastS ‘𝑈) ∈ 𝑉)
16 ccats1swrdeq 13402 . . . . 5 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))
1716imp 445 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)) → 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩))
1815, 17jca 554 . . 3 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)) → (( lastS ‘𝑈) ∈ 𝑉𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))
19 s1eq 13314 . . . . . 6 (𝑠 = ( lastS ‘𝑈) → ⟨“𝑠”⟩ = ⟨“( lastS ‘𝑈)”⟩)
2019oveq2d 6621 . . . . 5 (𝑠 = ( lastS ‘𝑈) → (𝑊 ++ ⟨“𝑠”⟩) = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩))
2120eqeq2d 2636 . . . 4 (𝑠 = ( lastS ‘𝑈) → (𝑈 = (𝑊 ++ ⟨“𝑠”⟩) ↔ 𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)))
2221rspcev 3300 . . 3 ((( lastS ‘𝑈) ∈ 𝑉𝑈 = (𝑊 ++ ⟨“( lastS ‘𝑈)”⟩)) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩))
2318, 22syl 17 . 2 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) ∧ 𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩)) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩))
2423ex 450 1 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ (#‘𝑈) = ((#‘𝑊) + 1)) → (𝑊 = (𝑈 substr ⟨0, (#‘𝑊)⟩) → ∃𝑠𝑉 𝑈 = (𝑊 ++ ⟨“𝑠”⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wrex 2913  c0 3896  cop 4159   class class class wbr 4618  cfv 5850  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884   < clt 10019  0cn0 11237  #chash 13054  Word cword 13225   lastS clsw 13226   ++ 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:  reuccats1lem  13412
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