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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 13870 | . . 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 4984 | . . . 4 ⊢ ∅ ∈ V | |
3 | ovex 6910 | . . . 4 ⊢ ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤)))) ∈ V | |
4 | 2, 3 | ifex 4325 | . . 3 ⊢ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) ∈ V |
5 | 1, 4 | dmmpt2 7476 | . 2 ⊢ dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) |
6 | id 22 | . . 3 ⊢ (¬ 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ) | |
7 | 6 | intnand 483 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) |
8 | ndmovg 7051 | . 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 582 | 1 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 {cab 2785 ∃wrex 3090 ∅c0 4115 ifcif 4277 〈cop 4374 × cxp 5310 dom cdm 5312 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 0cc0 10224 ℕ0cn0 11580 ℤcz 11666 ..^cfzo 12720 mod cmo 12923 ♯chash 13370 ++ cconcat 13590 substr csubstr 13664 prefix cpfx 13713 cyclShift ccsh 13868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-csh 13870 |
This theorem is referenced by: 0csh0 13877 cshwcl 13883 |
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