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| Mirrors > Home > MPE Home > Th. List > cshnz | Structured version Visualization version GIF version | ||
| Description: A cyclical shift is the empty set if the number of shifts is not an integer. (Contributed by Alexander van der Vekens, 21-May-2018.) (Revised by AV, 17-Nov-2018.) |
| Ref | Expression |
|---|---|
| cshnz | ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-csh 14695 | . . 3 ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))))) | |
| 2 | 0ex 5246 | . . . 4 ⊢ ∅ ∈ V | |
| 3 | ovex 7382 | . . . 4 ⊢ ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))) ∈ V | |
| 4 | 2, 3 | ifex 4527 | . . 3 ⊢ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) ∈ V |
| 5 | 1, 4 | dmmpo 8006 | . 2 ⊢ dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) |
| 6 | id 22 | . . 3 ⊢ (¬ 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ) | |
| 7 | 6 | intnand 488 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) |
| 8 | ndmovg 7532 | . 2 ⊢ ((dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) ∧ ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) → (𝑊 cyclShift 𝑁) = ∅) | |
| 9 | 5, 7, 8 | sylancr 587 | 1 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {cab 2707 ∃wrex 3053 ∅c0 4284 ifcif 4476 〈cop 4583 × cxp 5617 dom cdm 5619 Fn wfn 6477 ‘cfv 6482 (class class class)co 7349 0cc0 11009 ℕ0cn0 12384 ℤcz 12471 ..^cfzo 13557 mod cmo 13773 ♯chash 14237 ++ cconcat 14477 substr csubstr 14547 prefix cpfx 14577 cyclShift ccsh 14694 |
| 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-sep 5235 ax-nul 5245 ax-pr 5371 ax-un 7671 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-csh 14695 |
| This theorem is referenced by: 0csh0 14699 cshwcl 14704 |
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