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Theorem 2swrd1eqwrdeq 13306
 Description: Two (nonempty) words are equal if and only if they have the same prefix and the same single symbol suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)
Assertion
Ref Expression
2swrd1eqwrdeq ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))

Proof of Theorem 2swrd1eqwrdeq
StepHypRef Expression
1 lencl 13179 . . . . . . 7 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
2 nn0z 11277 . . . . . . 7 ((#‘𝑊) ∈ ℕ0 → (#‘𝑊) ∈ ℤ)
3 elnnz 11264 . . . . . . . 8 ((#‘𝑊) ∈ ℕ ↔ ((#‘𝑊) ∈ ℤ ∧ 0 < (#‘𝑊)))
43simplbi2 653 . . . . . . 7 ((#‘𝑊) ∈ ℤ → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ))
51, 2, 43syl 18 . . . . . 6 (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ))
65a1d 25 . . . . 5 (𝑊 ∈ Word 𝑉 → (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑊) → (#‘𝑊) ∈ ℕ)))
763imp 1249 . . . 4 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (#‘𝑊) ∈ ℕ)
8 fzo0end 12426 . . . 4 ((#‘𝑊) ∈ ℕ → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊)))
97, 8syl 17 . . 3 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊)))
10 2swrdeqwrdeq 13305 . . 3 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ ((#‘𝑊) − 1) ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)))))
119, 10syld3an3 1363 . 2 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)))))
12 hashneq0 13016 . . . . . . . . . . 11 (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
1312biimpd 218 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → (0 < (#‘𝑊) → 𝑊 ≠ ∅))
1413imdistani 722 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉𝑊 ≠ ∅))
15143adant2 1073 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 ∈ Word 𝑉𝑊 ≠ ∅))
1615adantr 480 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 ∈ Word 𝑉𝑊 ≠ ∅))
17 swrdlsw 13304 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)
1816, 17syl 17 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑊)”⟩)
19 breq2 4587 . . . . . . . . . 10 ((#‘𝑊) = (#‘𝑈) → (0 < (#‘𝑊) ↔ 0 < (#‘𝑈)))
20193anbi3d 1397 . . . . . . . . 9 ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ↔ (𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈))))
21 hashneq0 13016 . . . . . . . . . . . . 13 (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) ↔ 𝑈 ≠ ∅))
2221biimpd 218 . . . . . . . . . . . 12 (𝑈 ∈ Word 𝑉 → (0 < (#‘𝑈) → 𝑈 ≠ ∅))
2322imdistani 722 . . . . . . . . . . 11 ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉𝑈 ≠ ∅))
24233adant1 1072 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 ∈ Word 𝑉𝑈 ≠ ∅))
25 swrdlsw 13304 . . . . . . . . . 10 ((𝑈 ∈ Word 𝑉𝑈 ≠ ∅) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
2624, 25syl 17 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
2720, 26syl6bi 242 . . . . . . . 8 ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
2827impcom 445 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩)
29 oveq1 6556 . . . . . . . . . . 11 ((#‘𝑊) = (#‘𝑈) → ((#‘𝑊) − 1) = ((#‘𝑈) − 1))
30 id 22 . . . . . . . . . . 11 ((#‘𝑊) = (#‘𝑈) → (#‘𝑊) = (#‘𝑈))
3129, 30opeq12d 4348 . . . . . . . . . 10 ((#‘𝑊) = (#‘𝑈) → ⟨((#‘𝑊) − 1), (#‘𝑊)⟩ = ⟨((#‘𝑈) − 1), (#‘𝑈)⟩)
3231oveq2d 6565 . . . . . . . . 9 ((#‘𝑊) = (#‘𝑈) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩))
3332eqeq1d 2612 . . . . . . . 8 ((#‘𝑊) = (#‘𝑈) → ((𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
3433adantl 481 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩ ↔ (𝑈 substr ⟨((#‘𝑈) − 1), (#‘𝑈)⟩) = ⟨“( lastS ‘𝑈)”⟩))
3528, 34mpbird 246 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = ⟨“( lastS ‘𝑈)”⟩)
3618, 35eqeq12d 2625 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) ↔ ⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩))
37 hashgt0n0 13017 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → 𝑊 ≠ ∅)
38 lswcl 13208 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → ( lastS ‘𝑊) ∈ 𝑉)
3937, 38syldan 486 . . . . . . . 8 ((𝑊 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
40393adant2 1073 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑊) ∈ 𝑉)
4140adantr 480 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑊) ∈ 𝑉)
42 hashgt0n0 13017 . . . . . . . . . 10 ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → 𝑈 ≠ ∅)
43 lswcl 13208 . . . . . . . . . 10 ((𝑈 ∈ Word 𝑉𝑈 ≠ ∅) → ( lastS ‘𝑈) ∈ 𝑉)
4442, 43syldan 486 . . . . . . . . 9 ((𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
45443adant1 1072 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
4620, 45syl6bi 242 . . . . . . 7 ((#‘𝑊) = (#‘𝑈) → ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → ( lastS ‘𝑈) ∈ 𝑉))
4746impcom 445 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ( lastS ‘𝑈) ∈ 𝑉)
48 s111 13248 . . . . . 6 ((( lastS ‘𝑊) ∈ 𝑉 ∧ ( lastS ‘𝑈) ∈ 𝑉) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
4941, 47, 48syl2anc 691 . . . . 5 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (⟨“( lastS ‘𝑊)”⟩ = ⟨“( lastS ‘𝑈)”⟩ ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
5036, 49bitrd 267 . . . 4 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → ((𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) ↔ ( lastS ‘𝑊) = ( lastS ‘𝑈)))
5150anbi2d 736 . . 3 (((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) ∧ (#‘𝑊) = (#‘𝑈)) → (((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩)) ↔ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈))))
5251pm5.32da 671 . 2 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ (𝑊 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩) = (𝑈 substr ⟨((#‘𝑊) − 1), (#‘𝑊)⟩))) ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
5311, 52bitrd 267 1 ((𝑊 ∈ Word 𝑉𝑈 ∈ Word 𝑉 ∧ 0 < (#‘𝑊)) → (𝑊 = 𝑈 ↔ ((#‘𝑊) = (#‘𝑈) ∧ ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑈 substr ⟨0, ((#‘𝑊) − 1)⟩) ∧ ( lastS ‘𝑊) = ( lastS ‘𝑈)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   < clt 9953   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147  ⟨“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-fal 1481  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-s1 13157  df-substr 13158 This theorem is referenced by:  wwlkextinj  26258  clwwlkf1  26324  wwlksnextinj  41105  clwwlksf1  41224
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