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Mirrors > Home > MPE Home > Th. List > cshwsiun | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
cshwrepswhash1.m | ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} |
Ref | Expression |
---|---|
cshwsiun | ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3099 | . . 3 ⊢ {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} | |
2 | eqcom 2785 | . . . . . . . . 9 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 ↔ 𝑤 = (𝑊 cyclShift 𝑛)) | |
3 | 2 | biimpi 208 | . . . . . . . 8 ⊢ ((𝑊 cyclShift 𝑛) = 𝑤 → 𝑤 = (𝑊 cyclShift 𝑛)) |
4 | 3 | reximi 3192 | . . . . . . 7 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 → ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) |
5 | 4 | adantl 475 | . . . . . 6 ⊢ ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) → ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛)) |
6 | cshwcl 13949 | . . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑉 → (𝑊 cyclShift 𝑛) ∈ Word 𝑉) | |
7 | 6 | adantr 474 | . . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑛) ∈ Word 𝑉) |
8 | eleq1 2847 | . . . . . . . . 9 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ↔ (𝑊 cyclShift 𝑛) ∈ Word 𝑉)) | |
9 | 7, 8 | syl5ibrcom 239 | . . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0..^(♯‘𝑊))) → (𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉)) |
10 | 9 | rexlimdva 3213 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → 𝑤 ∈ Word 𝑉)) |
11 | eqcom 2785 | . . . . . . . . 9 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) ↔ (𝑊 cyclShift 𝑛) = 𝑤) | |
12 | 11 | biimpi 208 | . . . . . . . 8 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) → (𝑊 cyclShift 𝑛) = 𝑤) |
13 | 12 | reximi 3192 | . . . . . . 7 ⊢ (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) |
14 | 10, 13 | jca2 509 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) → (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤))) |
15 | 5, 14 | impbid2 218 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛))) |
16 | velsn 4414 | . . . . . . . 8 ⊢ (𝑤 ∈ {(𝑊 cyclShift 𝑛)} ↔ 𝑤 = (𝑊 cyclShift 𝑛)) | |
17 | 16 | bicomi 216 | . . . . . . 7 ⊢ (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)}) |
18 | 17 | a1i 11 | . . . . . 6 ⊢ (𝑊 ∈ Word 𝑉 → (𝑤 = (𝑊 cyclShift 𝑛) ↔ 𝑤 ∈ {(𝑊 cyclShift 𝑛)})) |
19 | 18 | rexbidv 3237 | . . . . 5 ⊢ (𝑊 ∈ Word 𝑉 → (∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 = (𝑊 cyclShift 𝑛) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)})) |
20 | 15, 19 | bitrd 271 | . . . 4 ⊢ (𝑊 ∈ Word 𝑉 → ((𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤) ↔ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)})) |
21 | 20 | abbidv 2906 | . . 3 ⊢ (𝑊 ∈ Word 𝑉 → {𝑤 ∣ (𝑤 ∈ Word 𝑉 ∧ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}}) |
22 | 1, 21 | syl5eq 2826 | . 2 ⊢ (𝑊 ∈ Word 𝑉 → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}}) |
23 | cshwrepswhash1.m | . 2 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} | |
24 | df-iun 4755 | . 2 ⊢ ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)} = {𝑤 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))𝑤 ∈ {(𝑊 cyclShift 𝑛)}} | |
25 | 22, 23, 24 | 3eqtr4g 2839 | 1 ⊢ (𝑊 ∈ Word 𝑉 → 𝑀 = ∪ 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {cab 2763 ∃wrex 3091 {crab 3094 {csn 4398 ∪ ciun 4753 ‘cfv 6135 (class class class)co 6922 0cc0 10272 ..^cfzo 12784 ♯chash 13435 Word cword 13599 cyclShift ccsh 13934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-substr 13731 df-pfx 13780 df-csh 13936 |
This theorem is referenced by: cshwsex 16206 cshwshashnsame 16209 |
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