Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pfxsuffeqwrdeq Structured version   Visualization version   GIF version

Theorem pfxsuffeqwrdeq 40269
 Description: Two words are equal if and only if they have the same prefix and the same suffix. Could replace 2swrdeqwrdeq 13305. (Contributed by AV, 5-May-2020.)
Assertion
Ref Expression
pfxsuffeqwrdeq ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))

Proof of Theorem pfxsuffeqwrdeq
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 eqwrd 13201 . . 3 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖))))
213adant3 1074 . 2 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖))))
3 elfzofz 12354 . . . . . . . . 9 (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ∈ (0...(#‘𝑊)))
4 fzosplit 12370 . . . . . . . . 9 (𝐼 ∈ (0...(#‘𝑊)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
53, 4syl 17 . . . . . . . 8 (𝐼 ∈ (0..^(#‘𝑊)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
653ad2ant3 1077 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
76adantr 480 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (0..^(#‘𝑊)) = ((0..^𝐼) ∪ (𝐼..^(#‘𝑊))))
87raleqdv 3121 . . . . 5 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖) ↔ ∀𝑖 ∈ ((0..^𝐼) ∪ (𝐼..^(#‘𝑊)))(𝑊𝑖) = (𝑆𝑖)))
9 ralunb 3756 . . . . 5 (∀𝑖 ∈ ((0..^𝐼) ∪ (𝐼..^(#‘𝑊)))(𝑊𝑖) = (𝑆𝑖) ↔ (∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖)))
108, 9syl6bb 275 . . . 4 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖) ↔ (∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖))))
11 eqidd 2611 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 = 𝐼)
1211biantrurd 528 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖))))
13 3simpa 1051 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉))
1413adantr 480 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉))
15 elfzonn0 12380 . . . . . . . . . 10 (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ∈ ℕ0)
1615, 15jca 553 . . . . . . . . 9 (𝐼 ∈ (0..^(#‘𝑊)) → (𝐼 ∈ ℕ0𝐼 ∈ ℕ0))
17163ad2ant3 1077 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝐼 ∈ ℕ0𝐼 ∈ ℕ0))
1817adantr 480 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ∈ ℕ0𝐼 ∈ ℕ0))
19 elfzo0le 12379 . . . . . . . . 9 (𝐼 ∈ (0..^(#‘𝑊)) → 𝐼 ≤ (#‘𝑊))
20193ad2ant3 1077 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → 𝐼 ≤ (#‘𝑊))
2120adantr 480 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 ≤ (#‘𝑊))
22 breq2 4587 . . . . . . . . . 10 ((#‘𝑆) = (#‘𝑊) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
2322eqcoms 2618 . . . . . . . . 9 ((#‘𝑊) = (#‘𝑆) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
2423adantl 481 . . . . . . . 8 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ≤ (#‘𝑆) ↔ 𝐼 ≤ (#‘𝑊)))
2521, 24mpbird 246 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → 𝐼 ≤ (#‘𝑆))
26 pfxeq 40267 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0𝐼 ∈ ℕ0) ∧ (𝐼 ≤ (#‘𝑊) ∧ 𝐼 ≤ (#‘𝑆))) → ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖))))
2714, 18, 21, 25, 26syl112anc 1322 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ↔ (𝐼 = 𝐼 ∧ ∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖))))
2812, 27bitr4d 270 . . . . 5 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖) ↔ (𝑊 prefix 𝐼) = (𝑆 prefix 𝐼)))
29 lencl 13179 . . . . . . . . . 10 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℕ0)
3029, 15anim12i 588 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → ((#‘𝑊) ∈ ℕ0𝐼 ∈ ℕ0))
31303adant2 1073 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → ((#‘𝑊) ∈ ℕ0𝐼 ∈ ℕ0))
3231ancomd 466 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0))
3332adantr 480 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0))
3429nn0red 11229 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ∈ ℝ)
3534leidd 10473 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (#‘𝑊) ≤ (#‘𝑊))
36353ad2ant1 1075 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (#‘𝑊) ≤ (#‘𝑊))
3736adantr 480 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑊))
38343ad2ant1 1075 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (#‘𝑊) ∈ ℝ)
39 eqle 10018 . . . . . . 7 (((#‘𝑊) ∈ ℝ ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑆))
4038, 39sylan 487 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (#‘𝑊) ≤ (#‘𝑆))
41 swrdspsleq 13301 . . . . . . 7 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) ∧ ((#‘𝑊) ≤ (#‘𝑊) ∧ (#‘𝑊) ≤ (#‘𝑆))) → ((𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩) ↔ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖)))
4241bicomd 212 . . . . . 6 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉) ∧ (𝐼 ∈ ℕ0 ∧ (#‘𝑊) ∈ ℕ0) ∧ ((#‘𝑊) ≤ (#‘𝑊) ∧ (#‘𝑊) ≤ (#‘𝑆))) → (∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖) ↔ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))
4314, 33, 37, 40, 42syl112anc 1322 . . . . 5 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖) ↔ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))
4428, 43anbi12d 743 . . . 4 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → ((∀𝑖 ∈ (0..^𝐼)(𝑊𝑖) = (𝑆𝑖) ∧ ∀𝑖 ∈ (𝐼..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖)) ↔ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩))))
4510, 44bitrd 267 . . 3 (((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) ∧ (#‘𝑊) = (#‘𝑆)) → (∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖) ↔ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩))))
4645pm5.32da 671 . 2 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (((#‘𝑊) = (#‘𝑆) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑆𝑖)) ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))
472, 46bitrd 267 1 ((𝑊 ∈ Word 𝑉𝑆 ∈ Word 𝑉𝐼 ∈ (0..^(#‘𝑊))) → (𝑊 = 𝑆 ↔ ((#‘𝑊) = (#‘𝑆) ∧ ((𝑊 prefix 𝐼) = (𝑆 prefix 𝐼) ∧ (𝑊 substr ⟨𝐼, (#‘𝑊)⟩) = (𝑆 substr ⟨𝐼, (#‘𝑊)⟩)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815   ≤ cle 9954  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150   prefix cpfx 40244 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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-substr 13158  df-pfx 40245 This theorem is referenced by:  pfxsuff1eqwrdeq  40270
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