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Theorem swrdrevpfx 35101
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 13672 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐿𝐹) ∈ (0...𝐿))
2 pfxcl 14712 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3 revcl 14796 . . . . . . . . 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 1132 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6 simp3 1137 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...𝐿))
7 revlen 14797 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = (♯‘(𝑊 prefix 𝐿)))
10 pfxlen 14718 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
119, 10eqtrd 2775 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12113adant3 1131 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
1312oveq2d 7447 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))) = (0...𝐿))
146, 13eleqtrrd 2842 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿)))))
155, 14jca 511 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
161, 15syl3an3 1164 . . . . 5 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐹 ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
17163com23 1125 . . . 4 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
18 revpfxsfxrev 35100 . . . 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 14802 . . . . . 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 7446 . . . 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 1132 . . 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 7446 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = (𝐿 − (𝐿𝐹)))
25243adant2 1130 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = (𝐿 − (𝐿𝐹)))
26 elfzel2 13559 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
2726zcnd 12721 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28 elfzelz 13561 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
2928zcnd 12721 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
3027, 29nncand 11623 . . . . . . 7 (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿𝐹)) = 𝐹)
31303ad2ant2 1133 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿𝐹)) = 𝐹)
3225, 31eqtrd 2775 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = 𝐹)
33113adant2 1130 . . . . 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 7447 . . 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 2779 . 2 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37 elfzuz3 13558 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ𝐹))
38 eluzfz2 13569 . . . . . . . 8 (𝐿 ∈ (ℤ𝐹) → 𝐿 ∈ (𝐹...𝐿))
3937, 38syl 17 . . . . . . 7 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (𝐹...𝐿))
4039ancli 548 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
41403ad2ant2 1133 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42 swrdpfx 14742 . . . . 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 1130 . . 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 2776 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 395  w3a 1086   = wceq 1537  wcel 2106  cop 4637  cfv 6563  (class class class)co 7431  0cc0 11153  cmin 11490  cuz 12876  ...cfz 13544  chash 14366  Word cword 14549   substr csubstr 14675   prefix cpfx 14705  reversecreverse 14793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-substr 14676  df-pfx 14706  df-reverse 14794
This theorem is referenced by:  swrdwlk  35111
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