Theorem cshwcshid 14958

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 30594 and erclwwlknsym 30643. (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 13749	. . . . . . 7 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2		oveq2 7420	. . . . . . . 8 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
3	2	eleq2d 2847	. . . . . . 7 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
4	1, 3	imbitrrid 249	. . . . . 6 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
5		cshwcshid.2	. . . . . 6 ⊢ (𝜑 → (♯‘𝑥) = (♯‘𝑦))
6	4, 5	syl11 34	. . . . 5 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
7	6	adantr 486	. . . 4 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
8	7	impcom 413	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9		cshwcshid.1	. . . . . . . 8 ⊢ (𝜑 → 𝑦 ∈ Word 𝑉)
10		simpl 488	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11		elfzelz 13637	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13		elfz2nn0 13732	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)))
14		nn0z 12698	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15		nn0z 12698	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
16		zsubcl 12719	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
17	14, 15, 16	syl2anr 609	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18	17	3adant3 1150	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
19	13, 18	sylbi 220	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
20	19	adantl 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
21	10, 12, 20	3jca 1146	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
22	9, 21	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23		2cshw 14944	. . . . . . 7 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
24	22, 23	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)) = (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))))
25		nn0cn 12597	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
26		nn0cn 12597	. . . . . . . . . . . 12 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
27	25, 26	anim12i 625	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28	27	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
29	13, 28	sylbi 220	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30		pncan3 11546	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
31	29, 30	syl 18	. . . . . . . 8 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
32	31	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
33	32	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34		cshwn 14928	. . . . . . . 8 ⊢ (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
35	9, 34	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
36	35	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
37	24, 33, 36	3eqtrrd 2801	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
38	37	adantrr 730	. . . 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)))
40	39	eqeq2d 2772	. . . . . 6 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
41	40	adantl 487	. . . . 5 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
42	41	adantl 487	. . . 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 260	. . 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)))
45	44	rspceeqv 3599	. . 3 ⊢ ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
46	8, 43, 45	syl2anc 596	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
47	46	ex 418	1 ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  ℂcc 11179  0cc0 11181   + caddc 11184   ≤ cle 11325   − cmin 11522  ℕ₀cn0 12587  ℤcz 12674  ...cfz 13620  ♯chash 14454  Word cword 14638   cyclShift ccsh 14919

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-rp 13102  df-fz 13621  df-fzo 13769  df-fl 13912  df-mod 13990  df-hash 14455  df-word 14639  df-concat 14696  df-substr 14769  df-pfx 14801  df-csh 14920

This theorem is used by:  erclwwlksym  30594  erclwwlknsym  30643