Theorem cshwcshid 13573

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlkssym 26935 and erclwwlksnsym 26947. (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 12446	. . . . . . 7 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦)))
2		oveq2 6658	. . . . . . . 8 ⊢ ((#‘𝑥) = (#‘𝑦) → (0...(#‘𝑥)) = (0...(#‘𝑦)))
3	2	eleq2d 2687	. . . . . . 7 ⊢ ((#‘𝑥) = (#‘𝑦) → (((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ↔ ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑦))))
4	1, 3	syl5ibr 236	. . . . . 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 481	. . . 4 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥))))
8	7	impcom 446	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)))
9		cshwcshid.1	. . . . . . . 8 ⊢ (𝜑 → 𝑦 ∈ Word 𝑉)
10		simpl 473	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(#‘𝑦))) → 𝑦 ∈ Word 𝑉)
11		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → 𝑚 ∈ ℤ)
12	11	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(#‘𝑦))) → 𝑚 ∈ ℤ)
13		elfz2nn0 12431	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(#‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (#‘𝑦)))
14		nn0z 11400	. . . . . . . . . . . . 13 ⊢ ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℤ)
15		nn0z 11400	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
16		zsubcl 11419	. . . . . . . . . . . . 13 ⊢ (((#‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((#‘𝑦) − 𝑚) ∈ ℤ)
17	14, 15, 16	syl2anr 495	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → ((#‘𝑦) − 𝑚) ∈ ℤ)
18	17	3adant3 1081	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
19	13, 18	sylbi 207	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → ((#‘𝑦) − 𝑚) ∈ ℤ)
20	19	adantl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(#‘𝑦))) → ((#‘𝑦) − 𝑚) ∈ ℤ)
21	10, 12, 20	3jca 1242	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
22	9, 21	sylan 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((#‘𝑦) − 𝑚) ∈ ℤ))
23		2cshw 13559	. . . . . . 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 11302	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
26		nn0cn 11302	. . . . . . . . . . . 12 ⊢ ((#‘𝑦) ∈ ℕ0 → (#‘𝑦) ∈ ℂ)
27	25, 26	anim12i 590	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
28	27	3adant3 1081	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ0 ∧ (#‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
29	13, 28	sylbi 207	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(#‘𝑦)) → (𝑚 ∈ ℂ ∧ (#‘𝑦) ∈ ℂ))
30		pncan3 10289	. . . . . . . . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑚 + ((#‘𝑦) − 𝑚)) = (#‘𝑦))
33	32	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (𝑚 + ((#‘𝑦) − 𝑚))) = (𝑦 cyclShift (#‘𝑦)))
34		cshwn 13543	. . . . . . . 8 ⊢ (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
35	9, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
36	35	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(#‘𝑦))) → (𝑦 cyclShift (#‘𝑦)) = 𝑦)
37	24, 33, 36	3eqtrrd 2661	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(#‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
38	37	adantrr 753	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
39		oveq1 6657	. . . . . . 7 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((#‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚)))
40	39	eqeq2d 2632	. . . . . 6 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
41	40	adantl 482	. . . . 5 ⊢ ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((#‘𝑦) − 𝑚))))
42	41	adantl 482	. . . 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 247	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
44		oveq2 6658	. . . . 5 ⊢ (𝑛 = ((#‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((#‘𝑦) − 𝑚)))
45	44	eqeq2d 2632	. . . 4 ⊢ (𝑛 = ((#‘𝑦) − 𝑚) → (𝑦 = (𝑥 cyclShift 𝑛) ↔ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))))
46	45	rspcev 3309	. . 3 ⊢ ((((#‘𝑦) − 𝑚) ∈ (0...(#‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((#‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
47	8, 43, 46	syl2anc 693	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
48	47	ex 450	1 ⊢ (𝜑 → ((𝑚 ∈ (0...(#‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(#‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936   + caddc 9939   ≤ cle 10075   − cmin 10266  ℕ₀cn0 11292  ℤcz 11377  ...cfz 12326  #chash 13117  Word cword 13291   cyclShift ccsh 13534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535

This theorem is referenced by:  erclwwlkssym  26935  erclwwlksnsym  26947