Theorem cshwcshid 14778

Description: A cyclically shifted word can be reconstructed by cyclically shifting it again. Lemma for erclwwlksym 29274 and erclwwlknsym 29323. (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 13608	. . . . . . 7 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦)))
2		oveq2 7417	. . . . . . . 8 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (0...(♯‘𝑥)) = (0...(♯‘𝑦)))
3	2	eleq2d 2820	. . . . . . 7 ⊢ ((♯‘𝑥) = (♯‘𝑦) → (((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ↔ ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑦))))
4	1, 3	imbitrrid 245	. . . . . 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 482	. . . 4 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝜑 → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥))))
8	7	impcom 409	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)))
9		cshwcshid.1	. . . . . . . 8 ⊢ (𝜑 → 𝑦 ∈ Word 𝑉)
10		simpl 484	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 ∈ Word 𝑉)
11		elfzelz 13501	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → 𝑚 ∈ ℤ)
12	11	adantl 483	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑚 ∈ ℤ)
13		elfz2nn0 13592	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) ↔ (𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)))
14		nn0z 12583	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℤ)
15		nn0z 12583	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
16		zsubcl 12604	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑦) ∈ ℤ ∧ 𝑚 ∈ ℤ) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
17	14, 15, 16	syl2anr 598	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
18	17	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
19	13, 18	sylbi 216	. . . . . . . . . 10 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
20	19	adantl 483	. . . . . . . . 9 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → ((♯‘𝑦) − 𝑚) ∈ ℤ)
21	10, 12, 20	3jca 1129	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
22	9, 21	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 ∈ Word 𝑉 ∧ 𝑚 ∈ ℤ ∧ ((♯‘𝑦) − 𝑚) ∈ ℤ))
23		2cshw 14763	. . . . . . 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 12482	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
26		nn0cn 12482	. . . . . . . . . . . 12 ⊢ ((♯‘𝑦) ∈ ℕ0 → (♯‘𝑦) ∈ ℂ)
27	25, 26	anim12i 614	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
28	27	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℕ0 ∧ (♯‘𝑦) ∈ ℕ0 ∧ 𝑚 ≤ (♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
29	13, 28	sylbi 216	. . . . . . . . 9 ⊢ (𝑚 ∈ (0...(♯‘𝑦)) → (𝑚 ∈ ℂ ∧ (♯‘𝑦) ∈ ℂ))
30		pncan3 11468	. . . . . . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑚 + ((♯‘𝑦) − 𝑚)) = (♯‘𝑦))
33	32	oveq2d 7425	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (𝑚 + ((♯‘𝑦) − 𝑚))) = (𝑦 cyclShift (♯‘𝑦)))
34		cshwn 14747	. . . . . . . 8 ⊢ (𝑦 ∈ Word 𝑉 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
35	9, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
36	35	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → (𝑦 cyclShift (♯‘𝑦)) = 𝑦)
37	24, 33, 36	3eqtrrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (0...(♯‘𝑦))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
38	37	adantrr 716	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
39		oveq1 7416	. . . . . . 7 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚)))
40	39	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = (𝑦 cyclShift 𝑚) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
41	40	adantl 483	. . . . 5 ⊢ ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → (𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)) ↔ 𝑦 = ((𝑦 cyclShift 𝑚) cyclShift ((♯‘𝑦) − 𝑚))))
42	41	adantl 483	. . . 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 7417	. . . 4 ⊢ (𝑛 = ((♯‘𝑦) − 𝑚) → (𝑥 cyclShift 𝑛) = (𝑥 cyclShift ((♯‘𝑦) − 𝑚)))
45	44	rspceeqv 3634	. . 3 ⊢ ((((♯‘𝑦) − 𝑚) ∈ (0...(♯‘𝑥)) ∧ 𝑦 = (𝑥 cyclShift ((♯‘𝑦) − 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
46	8, 43, 45	syl2anc 585	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚))) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛))
47	46	ex 414	1 ⊢ (𝜑 → ((𝑚 ∈ (0...(♯‘𝑦)) ∧ 𝑥 = (𝑦 cyclShift 𝑚)) → ∃𝑛 ∈ (0...(♯‘𝑥))𝑦 = (𝑥 cyclShift 𝑛)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  ℂcc 11108  0cc0 11110   + caddc 11113   ≤ cle 11249   − cmin 11444  ℕ₀cn0 12472  ℤcz 12558  ...cfz 13484  ♯chash 14290  Word cword 14464   cyclShift ccsh 14738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485  df-fzo 13628  df-fl 13757  df-mod 13835  df-hash 14291  df-word 14465  df-concat 14521  df-substr 14591  df-pfx 14621  df-csh 14739

This theorem is referenced by:  erclwwlksym  29274  erclwwlknsym  29323