Theorem cshwcshid 14793

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29950 and erclwwlknsym 29999. (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 13596	. . . . . . 7 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2		oveq2 7395	. . . . . . . 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 13485	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
12	11	adantl 481	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13		elfz2nn0 13579	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)))
14		nn0z 12554	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15		nn0z 12554	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
16		zsubcl 12575	. . . . . . . . . . . . 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 14778	. . . . . . 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 12452	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
26		nn0cn 12452	. . . . . . . . . . . 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 11429	. . . . . . . . 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 7403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34		cshwn 14762	. . . . . . . 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 7394	. . . . . . 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 7395	. . . 4 ⊢ (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
45	44	rspceeqv 3611	. . 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 5107  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  0cc0 11068   + caddc 11071   ≤ cle 11209   − cmin 11405  ℕ₀cn0 12442  ℤcz 12529  ...cfz 13468  ♯chash 14295  Word cword 14478   cyclShift ccsh 14753

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-fz 13469  df-fzo 13616  df-fl 13754  df-mod 13832  df-hash 14296  df-word 14479  df-concat 14536  df-substr 14606  df-pfx 14636  df-csh 14754

This theorem is referenced by:  erclwwlksym  29950  erclwwlknsym  29999