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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 13386 | . . 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 4718 | . . . 4 ⊢ ∅ ∈ V | |
3 | ovex 6577 | . . . 4 ⊢ ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉)) ∈ V | |
4 | 2, 3 | ifex 4106 | . . 3 ⊢ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉))) ∈ V |
5 | 1, 4 | dmmpt2 7129 | . 2 ⊢ dom cyclShift = ({𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} × ℤ) |
6 | id 22 | . . 3 ⊢ (¬ 𝑁 ∈ ℤ → ¬ 𝑁 ∈ ℤ) | |
7 | 6 | intnand 953 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → ¬ (𝑊 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ∧ 𝑁 ∈ ℤ)) |
8 | ndmovg 6715 | . 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 694 | 1 ⊢ (¬ 𝑁 ∈ ℤ → (𝑊 cyclShift 𝑁) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 ∅c0 3874 ifcif 4036 〈cop 4131 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ℤcz 11254 ..^cfzo 12334 mod cmo 12530 #chash 12979 ++ cconcat 13148 substr csubstr 13150 cyclShift ccsh 13385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-csh 13386 |
This theorem is referenced by: 0csh0 13390 cshwcl 13395 |
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