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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 14153 | . . . 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 4475 | . . . 4 ⊢ (𝑤 = ∅ → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) | |
4 | 3 | ad2antrl 726 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑤 = ∅ ∧ 𝑛 = 𝑁)) → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (♯‘𝑤)), (♯‘𝑤)〉) ++ (𝑤 prefix (𝑛 mod (♯‘𝑤))))) = ∅) |
5 | 0nn0 11915 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | f0 6562 | . . . . . . 7 ⊢ ∅:∅⟶V | |
7 | ffn 6516 | . . . . . . . 8 ⊢ (∅:∅⟶V → ∅ Fn ∅) | |
8 | fzo0 13064 | . . . . . . . . . 10 ⊢ (0..^0) = ∅ | |
9 | 8 | eqcomi 2832 | . . . . . . . . 9 ⊢ ∅ = (0..^0) |
10 | 9 | fneq2i 6453 | . . . . . . . 8 ⊢ (∅ Fn ∅ ↔ ∅ Fn (0..^0)) |
11 | 7, 10 | sylib 220 | . . . . . . 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 7166 | . . . . . . . . 9 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
15 | 14 | fneq2d 6449 | . . . . . . . 8 ⊢ (𝑙 = 0 → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
16 | 15 | adantl 484 | . . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑙 = 0) → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
17 | 13, 16 | rspcedv 3618 | . . . . . 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 5213 | . . . . . 6 ⊢ ∅ ∈ V | |
20 | fneq1 6446 | . . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓 Fn (0..^𝑙) ↔ ∅ Fn (0..^𝑙))) | |
21 | 20 | rexbidv 3299 | . . . . . 6 ⊢ (𝑓 = ∅ → (∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
22 | 19, 21 | elab 3669 | . . . . 5 ⊢ (∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙)) |
23 | 18, 22 | mpbir 233 | . . . 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 7304 | . 2 ⊢ (𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) |
28 | cshnz 14156 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) | |
29 | 27, 28 | pm2.61i 184 | 1 ⊢ (∅ cyclShift 𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 ∅c0 4293 ifcif 4469 〈cop 4575 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 0cc0 10539 ℕ0cn0 11900 ℤcz 11984 ..^cfzo 13036 mod cmo 13240 ♯chash 13693 ++ cconcat 13924 substr csubstr 14004 prefix cpfx 14034 cyclShift ccsh 14152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-csh 14153 |
This theorem is referenced by: cshw0 14158 cshwmodn 14159 cshwn 14161 cshwlen 14163 repswcshw 14176 |
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