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Mirrors > Home > MPE Home > Th. List > 0csh0 | Structured version Visualization version GIF version |
Description: Cyclically shifting an empty set/word always results in the empty word/set. (Contributed by AV, 25-Oct-2018.) (Revised by AV, 17-Nov-2018.) |
Ref | Expression |
---|---|
0csh0 | ⊢ (∅ cyclShift 𝑁) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csh 14824 | . . . 4 ⊢ 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 | 1 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → cyclShift = (𝑤 ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}, 𝑛 ∈ ℤ ↦ if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))))) |
3 | iftrue 4537 | . . . 4 ⊢ (𝑤 = ∅ → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) | |
4 | 3 | ad2antrl 728 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑤 = ∅ ∧ 𝑛 = 𝑁)) → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) |
5 | 0nn0 12539 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | f0 6790 | . . . . . . 7 ⊢ ∅:∅⟶V | |
7 | ffn 6737 | . . . . . . . 8 ⊢ (∅:∅⟶V → ∅ Fn ∅) | |
8 | fzo0 13720 | . . . . . . . . . 10 ⊢ (0..^0) = ∅ | |
9 | 8 | eqcomi 2744 | . . . . . . . . 9 ⊢ ∅ = (0..^0) |
10 | 9 | fneq2i 6667 | . . . . . . . 8 ⊢ (∅ Fn ∅ ↔ ∅ Fn (0..^0)) |
11 | 7, 10 | sylib 218 | . . . . . . 7 ⊢ (∅:∅⟶V → ∅ Fn (0..^0)) |
12 | 6, 11 | ax-mp 5 | . . . . . 6 ⊢ ∅ Fn (0..^0) |
13 | id 22 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 0 ∈ ℕ0) | |
14 | oveq2 7439 | . . . . . . . . 9 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
15 | 14 | fneq2d 6663 | . . . . . . . 8 ⊢ (𝑙 = 0 → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
16 | 15 | adantl 481 | . . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑙 = 0) → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
17 | 13, 16 | rspcedv 3615 | . . . . . 6 ⊢ (0 ∈ ℕ0 → (∅ Fn (0..^0) → ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
18 | 5, 12, 17 | mp2 9 | . . . . 5 ⊢ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙) |
19 | 0ex 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
20 | fneq1 6660 | . . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓 Fn (0..^𝑙) ↔ ∅ Fn (0..^𝑙))) | |
21 | 20 | rexbidv 3177 | . . . . . 6 ⊢ (𝑓 = ∅ → (∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
22 | 19, 21 | elab 3681 | . . . . 5 ⊢ (∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙)) |
23 | 18, 22 | mpbir 231 | . . . 4 ⊢ ∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} |
24 | 23 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → ∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)}) |
25 | id 22 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℤ) | |
26 | 19 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℤ → ∅ ∈ V) |
27 | 2, 4, 24, 25, 26 | ovmpod 7585 | . 2 ⊢ (𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) |
28 | cshnz 14827 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) | |
29 | 27, 28 | pm2.61i 182 | 1 ⊢ (∅ cyclShift 𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 {cab 2712 ∃wrex 3068 Vcvv 3478 ∅c0 4339 ifcif 4531 〈cop 4637 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 ℕ0cn0 12524 ℤcz 12611 ..^cfzo 13691 mod cmo 13906 ♯chash 14366 ++ cconcat 14605 substr csubstr 14675 prefix cpfx 14705 cyclShift ccsh 14823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-csh 14824 |
This theorem is referenced by: cshw0 14829 cshwmodn 14830 cshwn 14832 cshwlen 14834 repswcshw 14847 |
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