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Theorem swrdrevpfx 34759
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 13648 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐿𝐹) ∈ (0...𝐿))
2 pfxcl 14667 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3 revcl 14751 . . . . . . . . 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 14752 . . . . . . . . . . . . 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 479 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
10 pfxlen 14673 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
119, 10eqtrd 2768 . . . . . . . . . 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 7442 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
146, 13eleqtrrd 2832 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))))
155, 14jca 510 . . . . . 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 34758 . . . 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 14757 . . . . . 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 7441 . . . 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 7441 . . . . . . 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 13539 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
2726zcnd 12705 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28 elfzelz 13541 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
2928zcnd 12705 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
3027, 29nncand 11614 . . . . . . 7 (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿𝐹)) = 𝐹)
31303ad2ant2 1131 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿𝐹)) = 𝐹)
3225, 31eqtrd 2768 . . . . 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 4886 . . . 4 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ⟨((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)), (♯‘(reverse‘(𝑊 prefix 𝐿)))⟩ = ⟨𝐹, 𝐿⟩)
3534oveq2d 7442 . . 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 2772 . 2 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37 elfzuz3 13538 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ𝐹))
38 eluzfz2 13549 . . . . . . . 8 (𝐿 ∈ (ℤ𝐹) → 𝐿 ∈ (𝐹...𝐿))
3937, 38syl 17 . . . . . . 7 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (𝐹...𝐿))
4039ancli 547 . . . . . 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 14697 . . . . 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 2769 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 394  w3a 1084   = wceq 1533  wcel 2098  cop 4638  cfv 6553  (class class class)co 7426  0cc0 11146  cmin 11482  cuz 12860  ...cfz 13524  chash 14329  Word cword 14504   substr csubstr 14630   prefix cpfx 14660  reversecreverse 14748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-fzo 13668  df-hash 14330  df-word 14505  df-substr 14631  df-pfx 14661  df-reverse 14749
This theorem is referenced by:  swrdwlk  34769
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