Theorem cshwcshid 14752

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29983 and erclwwlknsym 30032. (Contributed by AV, 8-Apr-2018.) (Revised by AV, 11-Jun-2018.) (Proof shortened by AV, 3-Nov-2018.)

Hypotheses

Ref	Expression
cshwcshid.1	⊢ (𝜑 → 𝑦 ∈ 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

Step	Hyp	Ref	Expression
1		fznn0sub2 13556	. . . . . . 7 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2		oveq2 7361	. . . . . . . 8 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
3	2	eleq2d 2814	. . . . . . 7 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
4	1, 3	imbitrrid 246	. . . . . 6 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
5		cshwcshid.2	. . . . . 6 ⊢ (𝜑 → (♯‘𝑥) = (♯‘𝑦))
6	4, 5	syl11 33	. . . . 5 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
7	6	adantr 480	. . . 4 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
8	7	impcom 407	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9		cshwcshid.1	. . . . . . . 8 ⊢ (𝜑 → 𝑦 ∈ Word 𝑉)
10		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11		elfzelz 13445	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
12	11	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13		elfz2nn0 13539	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)))
14		nn0z 12514	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15		nn0z 12514	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
16		zsubcl 12535	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
17	14, 15, 16	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18	17	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
19	13, 18	sylbi 217	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
20	19	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
21	10, 12, 20	3jca 1128	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
22	9, 21	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23		2cshw 14737	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
24	22, 23	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
25		nn0cn 12412	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
26		nn0cn 12412	. . . . . . . . . . . 12 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
27	25, 26	anim12i 613	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28	27	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
29	13, 28	sylbi 217	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30		pncan3 11389	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
32	31	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
33	32	oveq2d 7369	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34		cshwn 14721	. . . . . . . 8 ⊢ (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
35	9, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
37	24, 33, 36	3eqtrrd 2769	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
38	37	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
39		oveq1 7360	. . . . . . 7 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
40	39	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
41	40	adantl 481	. . . . 5 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
42	41	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
43	38, 42	mpbird 257	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
44		oveq2 7361	. . . 4 ⊢ (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
45	44	rspceeqv 3602	. . 3 ⊢ ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
46	8, 43, 45	syl2anc 584	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
47	46	ex 412	1 ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  ℂcc 11026  0cc0 11028   + caddc 11031   ≤ cle 11169   − cmin 11365  ℕ₀cn0 12402  ℤcz 12489  ...cfz 13428  ♯chash 14255  Word cword 14438   cyclShift ccsh 14712

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-inf 9352  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-fz 13429  df-fzo 13576  df-fl 13714  df-mod 13792  df-hash 14256  df-word 14439  df-concat 14496  df-substr 14566  df-pfx 14596  df-csh 14713

This theorem is referenced by:  erclwwlksym  29983  erclwwlknsym  30032