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Theorem cshwcshid 14783
Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29542 and erclwwlknsym 29591. (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 fznn0sub2 13613 . . . . . . 7 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2 oveq2 7420 . . . . . . . 8 ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
32eleq2d 2818 . . . . . . 7 ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
41, 3imbitrrid 245 . . . . . 6 ((♯‘𝑥) = (♯‘𝑦) → (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
5 cshwcshid.2 . . . . . 6 (𝜑 → (♯‘𝑥) = (♯‘𝑦))
64, 5syl11 33 . . . . 5 (𝑚 ∈ (0...(♯‘𝑦)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
76adantr 480 . . . 4 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
87impcom 407 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9 cshwcshid.1 . . . . . . . 8 (𝜑𝑦 ∈ Word 𝑉)
10 simpl 482 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11 elfzelz 13506 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
1211adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13 elfz2nn0 13597 . . . . . . . . . . 11 (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)))
14 nn0z 12588 . . . . . . . . . . . . 13 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15 nn0z 12588 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ0𝑚 ∈ ℤ)
16 zsubcl 12609 . . . . . . . . . . . . 13 (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1714, 15, 16syl2anr 596 . . . . . . . . . . . 12 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18173adant3 1131 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
1913, 18sylbi 216 . . . . . . . . . 10 (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2019adantl 481 . . . . . . . . 9 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
2110, 12, 203jca 1127 . . . . . . . 8 ((𝑦 ∈ Word 𝑉𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
229, 21sylan 579 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23 2cshw 14768 . . . . . . 7 ((𝑦 ∈ Word 𝑉𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
2422, 23syl 17 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
25 nn0cn 12487 . . . . . . . . . . . 12 (𝑚 ∈ ℕ0𝑚 ∈ ℂ)
26 nn0cn 12487 . . . . . . . . . . . 12 ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
2725, 26anim12i 612 . . . . . . . . . . 11 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28273adant3 1131 . . . . . . . . . 10 ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
2913, 28sylbi 216 . . . . . . . . 9 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30 pncan3 11473 . . . . . . . . 9 ((𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3129, 30syl 17 . . . . . . . 8 (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3231adantl 481 . . . . . . 7 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
3332oveq2d 7428 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34 cshwn 14752 . . . . . . . 8 (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
359, 34syl 17 . . . . . . 7 (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3635adantr 480 . . . . . 6 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
3724, 33, 363eqtrrd 2776 . . . . 5 ((𝜑𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
3837adantrr 714 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
39 oveq1 7419 . . . . . . 7 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
4039eqeq2d 2742 . . . . . 6 (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4140adantl 481 . . . . 5 ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4241adantl 481 . . . 4 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
4338, 42mpbird 257 . . 3 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
44 oveq2 7420 . . . 4 (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
4544rspceeqv 3633 . . 3 ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
468, 43, 45syl2anc 583 . 2 ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
4746ex 412 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wrex 3069   class class class wbr 5148  cfv 6543  (class class class)co 7412  cc 11112  0cc0 11114   + caddc 11117  cle 11254  cmin 11449  0cn0 12477  cz 12563  ...cfz 13489  chash 14295  Word cword 14469   cyclShift ccsh 14743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-fl 13762  df-mod 13840  df-hash 14296  df-word 14470  df-concat 14526  df-substr 14596  df-pfx 14626  df-csh 14744
This theorem is referenced by:  erclwwlksym  29542  erclwwlknsym  29591
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