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Theorem swrdrevpfx 35122
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 13675 . . . . . 6 (𝐹 ∈ (0...𝐿) → (𝐿𝐹) ∈ (0...𝐿))
2 pfxcl 14715 . . . . . . . . 9 (𝑊 ∈ Word 𝑉 → (𝑊 prefix 𝐿) ∈ Word 𝑉)
3 revcl 14799 . . . . . . . . 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 1134 . . . . . . 7 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉)
6 simp3 1139 . . . . . . . 8 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ (𝐿𝐹) ∈ (0...𝐿)) → (𝐿𝐹) ∈ (0...𝐿))
7 revlen 14800 . . . . . . . . . . . . 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 14721 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐿)) = 𝐿)
119, 10eqtrd 2777 . . . . . . . . . 10 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
12113adant3 1133 . . . . . . . . 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 2844 . . . . . . 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 1166 . . . . 5 ((𝑊 ∈ Word 𝑉𝐿 ∈ (0...(♯‘𝑊)) ∧ 𝐹 ∈ (0...𝐿)) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
17163com23 1127 . . . 4 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((reverse‘(𝑊 prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿𝐹) ∈ (0...(♯‘(reverse‘(𝑊 prefix 𝐿))))))
18 revpfxsfxrev 35121 . . . 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 14805 . . . . . 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 1134 . . 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 1132 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = (𝐿 − (𝐿𝐹)))
26 elfzel2 13562 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
2726zcnd 12723 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ ℂ)
28 elfzelz 13564 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℤ)
2928zcnd 12723 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐹 ∈ ℂ)
3027, 29nncand 11625 . . . . . . 7 (𝐹 ∈ (0...𝐿) → (𝐿 − (𝐿𝐹)) = 𝐹)
31303ad2ant2 1135 . . . . . 6 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐿 − (𝐿𝐹)) = 𝐹)
3225, 31eqtrd 2777 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → ((♯‘(reverse‘(𝑊 prefix 𝐿))) − (𝐿𝐹)) = 𝐹)
33113adant2 1132 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (♯‘(reverse‘(𝑊 prefix 𝐿))) = 𝐿)
3432, 33opeq12d 4881 . . . 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 2781 . 2 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (reverse‘((reverse‘(𝑊 prefix 𝐿)) prefix (𝐿𝐹))) = ((𝑊 prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37 elfzuz3 13561 . . . . . . . 8 (𝐹 ∈ (0...𝐿) → 𝐿 ∈ (ℤ𝐹))
38 eluzfz2 13572 . . . . . . . 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 1135 . . . . 5 ((𝑊 ∈ Word 𝑉𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(♯‘𝑊))) → (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42 swrdpfx 14745 . . . . 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 1132 . . 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 2778 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 1087   = wceq 1540  wcel 2108  cop 4632  cfv 6561  (class class class)co 7431  0cc0 11155  cmin 11492  cuz 12878  ...cfz 13547  chash 14369  Word cword 14552   substr csubstr 14678   prefix cpfx 14708  reversecreverse 14796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-substr 14679  df-pfx 14709  df-reverse 14797
This theorem is referenced by:  swrdwlk  35132
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