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Theorem cshwcshid 13424
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlktr 26343 and erclwwlkntr 26355. (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 12315 . . . . . . . 8 (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦)))
3 oveq2 6557 . . . . . . . . 9 ((#‘𝑥) = (#‘𝑦) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
43eleq2d 2673 . . . . . . . 8 ((#‘𝑥) = (#‘𝑦) → (((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ↔ ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦))))
52, 4syl5ibr 235 . . . . . . 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 480 . . . 4 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
98impcom 445 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)))
10 cshwcshid.1 . . . . . . . 8 (𝜑𝑦 ∈ Word 𝑉)
11 simpl 472 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → 𝑦 ∈ Word 𝑉)
12 elfzelz 12213 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
1312adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → 𝑚 ∈ ℤ)
14 elfz2nn0 12300 . . . . . . . . . . 11 (𝑚 ∈ (0...(#‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)))
15 nn0z 11277 . . . . . . . . . . . . 13 ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℤ)
16 nn0z 11277 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
17 zsubcl 11296 . . . . . . . . . . . . 13 (((#‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((#‘𝑦) − 𝑚) ∈ ℤ)
1815, 16, 17syl2anr 494 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → ((#‘𝑦) − 𝑚) ∈ ℤ)
19183adant3 1074 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2014, 19sylbi 206 . . . . . . . . . 10 (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2120adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → ((#‘𝑦) − 𝑚) ∈ ℤ)
2211, 13, 213jca 1235 . . . . . . . 8 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
2310, 22sylan 487 . . . . . . 7 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
24 2cshw 13410 . . . . . . 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 11179 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
27 nn0cn 11179 . . . . . . . . . . . 12 ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℂ)
2826, 27anim12i 588 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
29283adant3 1074 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0𝑚 ≤ (#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
3014, 29sylbi 206 . . . . . . . . 9 (𝑚 ∈ (0...(#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
31 pncan3 10168 . . . . . . . . 9 ((𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3230, 31syl 17 . . . . . . . 8 (𝑚 ∈ (0...(#‘𝑦)) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3332adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
3433oveq2d 6565 . . . . . 6 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (𝑚 + ((#‘𝑦) − 𝑚))) = (𝑦 cyclShift (#‘𝑦)))
35 cshwn 13394 . . . . . . . 8 (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3610, 35syl 17 . . . . . . 7 (𝜑 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3736adantr 480 . . . . . 6 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
3825, 34, 373eqtrrd 2649 . . . . 5 ((𝜑𝑚 ∈ (0...(#‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
3938adantrr 749 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
40 oveq1 6556 . . . . . . 7 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((#‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
4140eqeq2d 2620 . . . . . 6 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4241adantl 481 . . . . 5 ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4342adantl 481 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
4439, 43mpbird 246 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
45 oveq2 6557 . . . . 5 (𝑛 = ((#‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
4645eqeq2d 2620 . . . 4 (𝑛 = ((#‘𝑦) − 𝑚) → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))))
4746rspcev 3282 . . 3 ((((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
489, 44, 47syl2anc 691 . 2 ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
4948ex 449 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wrex 2897   class class class wbr 4583  cfv 5804  (class class class)co 6549  cc 9813  0cc0 9815   + caddc 9818  cle 9954  cmin 10145  0cn0 11169  cz 11254  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386
This theorem is referenced by:  erclwwlksym  26342  erclwwlknsym  26354  erclwwlkssym  41242  erclwwlksnsym  41254
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