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Theorem swrdrevpfx 34782
Description: A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
Assertion
Ref Expression
swrdrevpfx ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨𝐹, 𝐿⟩) = (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))))

Proof of Theorem swrdrevpfx
StepHypRef Expression
1 fznn0sub2 13638 . . . . . 6 (𝐹 ∈ (0...𝐿) β†’ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿))
2 pfxcl 14657 . . . . . . . . 9 (π‘Š ∈ Word 𝑉 β†’ (π‘Š prefix 𝐿) ∈ Word 𝑉)
3 revcl 14741 . . . . . . . . 9 ((π‘Š prefix 𝐿) ∈ Word 𝑉 β†’ (reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉)
42, 3syl 17 . . . . . . . 8 (π‘Š ∈ Word 𝑉 β†’ (reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉)
543ad2ant1 1130 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉)
6 simp3 1135 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿))
7 revlen 14742 . . . . . . . . . . . . 13 ((π‘Š prefix 𝐿) ∈ Word 𝑉 β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
82, 7syl 17 . . . . . . . . . . . 12 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
98adantr 479 . . . . . . . . . . 11 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (β™―β€˜(π‘Š prefix 𝐿)))
10 pfxlen 14663 . . . . . . . . . . 11 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(π‘Š prefix 𝐿)) = 𝐿)
119, 10eqtrd 2765 . . . . . . . . . 10 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = 𝐿)
12113adant3 1129 . . . . . . . . 9 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = 𝐿)
1312oveq2d 7431 . . . . . . . 8 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))) = (0...𝐿))
146, 13eleqtrrd 2828 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ (𝐿 βˆ’ 𝐹) ∈ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))))
155, 14jca 510 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ (𝐿 βˆ’ 𝐹) ∈ (0...𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 βˆ’ 𝐹) ∈ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))))))
161, 15syl3an3 1162 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š)) ∧ 𝐹 ∈ (0...𝐿)) β†’ ((reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 βˆ’ 𝐹) ∈ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))))))
17163com23 1123 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 βˆ’ 𝐹) ∈ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))))))
18 revpfxsfxrev 34781 . . . 4 (((reverseβ€˜(π‘Š prefix 𝐿)) ∈ Word 𝑉 ∧ (𝐿 βˆ’ 𝐹) ∈ (0...(β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))))) β†’ (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))) = ((reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩))
1917, 18syl 17 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))) = ((reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩))
20 revrev 14747 . . . . . 6 ((π‘Š prefix 𝐿) ∈ Word 𝑉 β†’ (reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (π‘Š prefix 𝐿))
212, 20syl 17 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) = (π‘Š prefix 𝐿))
2221oveq1d 7430 . . . 4 (π‘Š ∈ Word 𝑉 β†’ ((reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩) = ((π‘Š prefix 𝐿) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩))
23223ad2ant1 1130 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((reverseβ€˜(reverseβ€˜(π‘Š prefix 𝐿))) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩) = ((π‘Š prefix 𝐿) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩))
2411oveq1d 7430 . . . . . . 7 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)) = (𝐿 βˆ’ (𝐿 βˆ’ 𝐹)))
25243adant2 1128 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)) = (𝐿 βˆ’ (𝐿 βˆ’ 𝐹)))
26 elfzel2 13529 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) β†’ 𝐿 ∈ β„€)
2726zcnd 12695 . . . . . . . 8 (𝐹 ∈ (0...𝐿) β†’ 𝐿 ∈ β„‚)
28 elfzelz 13531 . . . . . . . . 9 (𝐹 ∈ (0...𝐿) β†’ 𝐹 ∈ β„€)
2928zcnd 12695 . . . . . . . 8 (𝐹 ∈ (0...𝐿) β†’ 𝐹 ∈ β„‚)
3027, 29nncand 11604 . . . . . . 7 (𝐹 ∈ (0...𝐿) β†’ (𝐿 βˆ’ (𝐿 βˆ’ 𝐹)) = 𝐹)
31303ad2ant2 1131 . . . . . 6 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (𝐿 βˆ’ (𝐿 βˆ’ 𝐹)) = 𝐹)
3225, 31eqtrd 2765 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)) = 𝐹)
33113adant2 1128 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) = 𝐿)
3432, 33opeq12d 4877 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩ = ⟨𝐹, 𝐿⟩)
3534oveq2d 7431 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿) substr ⟨((β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿))) βˆ’ (𝐿 βˆ’ 𝐹)), (β™―β€˜(reverseβ€˜(π‘Š prefix 𝐿)))⟩) = ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
3619, 23, 353eqtrd 2769 . 2 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))) = ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩))
37 elfzuz3 13528 . . . . . . . 8 (𝐹 ∈ (0...𝐿) β†’ 𝐿 ∈ (β„€β‰₯β€˜πΉ))
38 eluzfz2 13539 . . . . . . . 8 (𝐿 ∈ (β„€β‰₯β€˜πΉ) β†’ 𝐿 ∈ (𝐹...𝐿))
3937, 38syl 17 . . . . . . 7 (𝐹 ∈ (0...𝐿) β†’ 𝐿 ∈ (𝐹...𝐿))
4039ancli 547 . . . . . 6 (𝐹 ∈ (0...𝐿) β†’ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
41403ad2ant2 1131 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)))
42 swrdpfx 14687 . . . . 5 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (𝐹...𝐿)) β†’ ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (π‘Š substr ⟨𝐹, 𝐿⟩)))
4341, 42syl5 34 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (π‘Š substr ⟨𝐹, 𝐿⟩)))
44433adant2 1128 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (π‘Š substr ⟨𝐹, 𝐿⟩)))
4544pm2.43i 52 . 2 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ ((π‘Š prefix 𝐿) substr ⟨𝐹, 𝐿⟩) = (π‘Š substr ⟨𝐹, 𝐿⟩))
4636, 45eqtr2d 2766 1 ((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨𝐹, 𝐿⟩) = (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4630  β€˜cfv 6542  (class class class)co 7415  0cc0 11136   βˆ’ cmin 11472  β„€β‰₯cuz 12850  ...cfz 13514  β™―chash 14319  Word cword 14494   substr csubstr 14620   prefix cpfx 14650  reversecreverse 14738
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-substr 14621  df-pfx 14651  df-reverse 14739
This theorem is referenced by:  swrdwlk  34792
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