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Theorem swrdrevpfx 32383
 Description: A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
Assertion
Ref Expression
swrdrevpfx ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))))

Proof of Theorem swrdrevpfx
StepHypRef Expression
1 fznn0sub2 13009 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐿𝐹) ∈ (0...𝐿))
2 pfxcl 14030 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3 revcl 14114 . . . . . . . . 9 ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
42, 3syl 17 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
543ad2ant1 1130 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6 simp3 1135 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...𝐿))
7 revlen 14115 . . . . . . . . . . . . 13 ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
82, 7syl 17 . . . . . . . . . . . 12 (𝑊 ∈ Word 𝑉 → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
98adantr 484 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
10 pfxlen 14036 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
119, 10eqtrd 2859 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12113adant3 1129 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
1312oveq2d 7156 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
146, 13eleqtrrd 2919 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))))
155, 14jca 515 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
161, 15syl3an3 1162 . . . . 5 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐹 ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
17163com23 1123 . . . 4 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
18 revpfxsfxrev 32382 . . . 4 (((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
1917, 18syl 17 . . 3 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
20 revrev 14120 . . . . . 6 ((𝑊 prefix 𝐿) ∈ Word 𝑉 → (reverse‘(reverse‘(𝑊 prefix 𝐿))) = (𝑊 prefix 𝐿))
212, 20syl 17 . . . . 5 (𝑊 ∈ Word 𝑉 → (reverse‘(reverse‘(𝑊 prefix 𝐿))) = (𝑊 prefix 𝐿))
2221oveq1d 7155 . . . 4 (𝑊 ∈ Word 𝑉 → ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
23223ad2ant1 1130 . . 3 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(reverse‘(𝑊 prefix 𝐿))) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩))
2411oveq1d 7155 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = (𝐿 − (𝐿𝐹)))
25243adant2 1128 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = (𝐿 − (𝐿𝐹)))
26 elfzel2 12900 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
2726zcnd 12076 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28 elfzelz 12902 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
2928zcnd 12076 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
3027, 29nncand 10989 . . . . . . 7 (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿𝐹)) = 𝐹)
31303ad2ant2 1131 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿𝐹)) = 𝐹)
3225, 31eqtrd 2859 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = 𝐹)
33113adant2 1128 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
3432, 33opeq12d 4794 . . . 4 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩ = ⟨𝐹, 𝐿⟩)
3534oveq2d 7156 . . 3 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
3619, 23, 353eqtrd 2863 . 2 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37 elfzuz3 12899 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ𝐹))
38 eluzfz2 12910 . . . . . . . 8 (𝐿 ∈ (ℤ𝐹) → 𝐿 ∈ (𝐹...𝐿))
3937, 38syl 17 . . . . . . 7 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (𝐹...𝐿))
4039ancli 552 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
41403ad2ant2 1131 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42 swrdpfx 14060 . . . . 5 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → ((𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
4341, 42syl5 34 . . . 4 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
44433adant2 1128 . . 3 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩)))
4544pm2.43i 52 . 2 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (𝑊 substr ⟨𝐹, 𝐿⟩))
4636, 45eqtr2d 2860 1 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝑊 substr ⟨𝐹, 𝐿⟩) = (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ⟨cop 4554  ‘cfv 6338  (class class class)co 7140  0cc0 10524   − cmin 10857  ℤ≥cuz 12231  ...cfz 12885  ♯chash 13686  Word cword 13857   substr csubstr 13993   prefix cpfx 14023  reversecreverse 14111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446  ax-cnex 10580  ax-resscn 10581  ax-1cn 10582  ax-icn 10583  ax-addcl 10584  ax-addrcl 10585  ax-mulcl 10586  ax-mulrcl 10587  ax-mulcom 10588  ax-addass 10589  ax-mulass 10590  ax-distr 10591  ax-i2m1 10592  ax-1ne0 10593  ax-1rid 10594  ax-rnegex 10595  ax-rrecex 10596  ax-cnre 10597  ax-pre-lttri 10598  ax-pre-lttrn 10599  ax-pre-ltadd 10600  ax-pre-mulgt0 10601 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-int 4860  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7674  df-2nd 7675  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-card 9354  df-pnf 10664  df-mnf 10665  df-xr 10666  df-ltxr 10667  df-le 10668  df-sub 10859  df-neg 10860  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-substr 13994  df-pfx 14024  df-reverse 14112 This theorem is referenced by:  swrdwlk  32393
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