Theorem cshwsiun 15806

Description: The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)

Hypothesis

Ref	Expression
cshwrepswhash1.m	⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}

Assertion

Ref	Expression
cshwsiun	⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)})

Distinct variable groups:   𝑛,𝑉,𝑤   𝑛,𝑊,𝑤

Allowed substitution hints:   𝑀(𝑤,𝑛)

Proof of Theorem cshwsiun

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)}
2		eqcom 2629	. . . . . . . . 9 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛))
3	2	biimpi 206	. . . . . . . 8 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 → 𝑤 = (𝑊 cyclShift 𝑛))
4	3	reximi 3011	. . . . . . 7 ⊢ (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 → ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
5	4	adantl 482	. . . . . 6 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) → ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))
6		cshwcl 13544	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑛) ∈ Word 𝑉)
8		eleq1 2689	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ↔ (𝑊 cyclShift 𝑛) ∈ Word 𝑉))
9	7, 8	syl5ibrcom 237	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0..^(#‘𝑊))) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉))
10	9	rexlimdva 3031	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉))
11	10	imp 445	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → 𝑤 ∈ Word 𝑉)
12		eqcom 2629	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) ↔ (𝑊 cyclShift 𝑛) = 𝑤)
13	12	biimpi 206	. . . . . . . . . 10 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑊 cyclShift 𝑛) = 𝑤)
14	13	reximi 3011	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
15	14	adantl 482	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)
16	11, 15	jca 554	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))
17	16	ex 450	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)))
18	5, 17	impbid2 216	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)))
19		velsn 4193	. . . . . . . 8 ⊢ (𝑤 ∈ {(𝑊 cyclShift 𝑛)} ↔ 𝑤 = (𝑊 cyclShift 𝑛))
20	19	bicomi 214	. . . . . . 7 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)})
21	20	a1i 11	. . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
22	21	rexbidv 3052	. . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(#‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
23	18, 22	bitrd 268	. . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}))
24	23	abbidv 2741	. . 3 ⊢ (𝑊 ∈ Word 𝑉 → {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}})
25	1, 24	syl5eq 2668	. 2 ⊢ (𝑊 ∈ Word 𝑉 → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}})
26		cshwrepswhash1.m	. 2 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
27		df-iun 4522	. 2 ⊢ ∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}}
28	25, 26, 27	3eqtr4g 2681	1 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(#‘𝑊)){(𝑊 cyclShift 𝑛)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  {crab 2916  {csn 4177  ∪ ciun 4520  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ..^cfzo 12465  #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

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-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-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535

This theorem is referenced by:  cshwsex  15807  cshwshashnsame  15810