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Theorem cshwcshid 13366
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlktr 26105 and erclwwlkntr 26117. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)
Hypotheses
Ref Expression
cshwcshid.1 (𝜑𝑦 ∈ Word 𝑉)
cshwcshid.2 (𝜑 → (#‘𝑥) = (#‘𝑦))
Assertion
Ref Expression
cshwcshid (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
Distinct variable group:   𝑚,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚,𝑛)   𝑉(𝑥,𝑦,𝑚,𝑛)

Proof of Theorem cshwcshid
StepHypRef Expression
1 cshwcshid.2 . . . . . . 7 (𝜑 → (#‘𝑥) = (#‘𝑦))
2 fznn0sub2 12266 . . . . . . . 8 (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦)))
3 oveq2 6531 . . . . . . . . 9 ((#‘𝑥) = (#‘𝑦) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
43eleq2d 2668 . . . . . . . 8 ((#‘𝑥) = (#‘𝑦) → (((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ↔ ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦))))
52, 4syl5ibr 234 . . . . . . 7 ((#‘𝑥) = (#‘𝑦) → (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
61, 5syl 17 . . . . . 6 (𝜑 → (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
76com12 32 . . . . 5 (𝑚 ∈ (0...(#‘𝑦)) → (𝜑 → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
87adantr 479 . . . 4 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
98impcom 444 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)))
10 cshwcshid.1 . . . . . . . 8 (𝜑𝑦 ∈ Word 𝑉)
11 simpl 471 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → 𝑦 ∈ Word 𝑉)
12 elfzelz 12164 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
1312adantl 480 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → 𝑚 ∈ ℤ)
14 elfz2nn0 12251 . . . . . . . . . . 11 (𝑚 ∈ (0...(#‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)))
15 nn0z 11229 . . . . . . . . . . . . 13 ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℤ)
16 nn0z 11229 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
17 zsubcl 11248 . . . . . . . . . . . . 13 (((#‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((#‘𝑦) − 𝑚) ∈ ℤ)
1815, 16, 17syl2anr 493 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → ((#‘𝑦) − 𝑚) ∈ ℤ)
19183adant3 1073 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2014, 19sylbi 205 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2120adantl 480 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2211, 13, 213jca 1234 . . . . . . . 8 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
2310, 22sylan 486 . . . . . . 7 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
24 2cshw 13352 . . . . . . 7 ((𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ) → ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((#‘𝑦) − 𝑚))))
2523, 24syl 17 . . . . . 6 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((#‘𝑦) − 𝑚))))
26 nn0cn 11145 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
27 nn0cn 11145 . . . . . . . . . . . 12 ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℂ)
2826, 27anim12i 587 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
29283adant3 1073 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
3014, 29sylbi 205 . . . . . . . . 9 (𝑚 ∈ (0...(#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
31 pncan3 10136 . . . . . . . . 9 ((𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3230, 31syl 17 . . . . . . . 8 (𝑚 ∈ (0...(#‘𝑦)) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3332adantl 480 . . . . . . 7 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3433oveq2d 6539 . . . . . 6 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (𝑚 + ((#‘𝑦) − 𝑚))) = (𝑦 cyclShift (#‘𝑦)))
35 cshwn 13336 . . . . . . . 8 (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3610, 35syl 17 . . . . . . 7 (𝜑 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3736adantr 479 . . . . . 6 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3825, 34, 373eqtrrd 2644 . . . . 5 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
3938adantrr 748 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
40 oveq1 6530 . . . . . . 7 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((#‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
4140eqeq2d 2615 . . . . . 6 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4241adantl 480 . . . . 5 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4342adantl 480 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4439, 43mpbird 245 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
45 oveq2 6531 . . . . 5 (𝑛 = ((#‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
4645eqeq2d 2615 . . . 4 (𝑛 = ((#‘𝑦) − 𝑚) → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))))
4746rspcev 3277 . . 3 ((((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
489, 44, 47syl2anc 690 . 2 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
4948ex 448 1 (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wrex 2892   class class class wbr 4573  cfv 5786  (class class class)co 6523  cc 9786  0cc0 9788   + caddc 9791  cle 9927  cmin 10113  0cn0 11135  cz 11206  ...cfz 12148  #chash 12930  Word cword 13088   cyclShift ccsh 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-csh 13328
This theorem is referenced by:  erclwwlksym  26104  erclwwlknsym  26116  erclwwlkssym  41240  erclwwlksnsym  41252
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