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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 13535 | . . 3 ⊢ cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉)))) | |
2 | 0ex 4790 | . . . 4 ⊢ ∅ ∈ V | |
3 | ovex 6678 | . . . 4 ⊢ ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉)) ∈ V | |
4 | 2, 3 | ifex 4156 | . . 3 ⊢ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉))) ∈ V |
5 | 1, 4 | dmmpt2 7240 | . 2 ⊢ dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) |
6 | id 22 | . . 3 ⊢ (¬ 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ) | |
7 | 6 | intnand 962 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) |
8 | ndmovg 6817 | . 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 695 | 1 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 ∅c0 3915 ifcif 4086 〈cop 4183 × cxp 5112 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ℕ0cn0 11292 ℤcz 11377 ..^cfzo 12465 mod cmo 12668 #chash 13117 ++ cconcat 13293 substr csubstr 13295 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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-csh 13535 |
This theorem is referenced by: 0csh0 13539 cshwcl 13544 |
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