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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 13535 | . . . 4 ⊢ 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 | 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 substr 〈0, (𝑛 mod (#‘𝑤))〉))))) |
3 | iftrue 4092 | . . . 4 ⊢ (𝑤 = ∅ → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉))) = ∅) | |
4 | 3 | ad2antrl 764 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑤 = ∅ ∧ 𝑛 = 𝑁)) → if(𝑤 = ∅, ∅, ((𝑤 substr 〈(𝑛 mod (#‘𝑤)), (#‘𝑤)〉) ++ (𝑤 substr 〈0, (𝑛 mod (#‘𝑤))〉))) = ∅) |
5 | 0nn0 11307 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
6 | f0 6086 | . . . . . . 7 ⊢ ∅:∅⟶V | |
7 | ffn 6045 | . . . . . . . 8 ⊢ (∅:∅⟶V → ∅ Fn ∅) | |
8 | fzo0 12492 | . . . . . . . . . 10 ⊢ (0..^0) = ∅ | |
9 | 8 | eqcomi 2631 | . . . . . . . . 9 ⊢ ∅ = (0..^0) |
10 | 9 | fneq2i 5986 | . . . . . . . 8 ⊢ (∅ Fn ∅ ↔ ∅ Fn (0..^0)) |
11 | 7, 10 | sylib 208 | . . . . . . 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 6658 | . . . . . . . . 9 ⊢ (𝑙 = 0 → (0..^𝑙) = (0..^0)) | |
15 | 14 | fneq2d 5982 | . . . . . . . 8 ⊢ (𝑙 = 0 → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
16 | 15 | adantl 482 | . . . . . . 7 ⊢ ((0 ∈ ℕ0 ∧ 𝑙 = 0) → (∅ Fn (0..^𝑙) ↔ ∅ Fn (0..^0))) |
17 | 13, 16 | rspcedv 3313 | . . . . . 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 4790 | . . . . . 6 ⊢ ∅ ∈ V | |
20 | fneq1 5979 | . . . . . . 7 ⊢ (𝑓 = ∅ → (𝑓 Fn (0..^𝑙) ↔ ∅ Fn (0..^𝑙))) | |
21 | 20 | rexbidv 3052 | . . . . . 6 ⊢ (𝑓 = ∅ → (∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙) ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙))) |
22 | 19, 21 | elab 3350 | . . . . 5 ⊢ (∅ ∈ {𝑓 ∣ ∃𝑙 ∈ ℕ0 𝑓 Fn (0..^𝑙)} ↔ ∃𝑙 ∈ ℕ0 ∅ Fn (0..^𝑙)) |
23 | 18, 22 | mpbir 221 | . . . 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 | ovmpt2d 6788 | . 2 ⊢ (𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) |
28 | cshnz 13538 | . 2 ⊢ (¬ 𝑁 ∈ ℤ → (∅ cyclShift 𝑁) = ∅) | |
29 | 27, 28 | pm2.61i 176 | 1 ⊢ (∅ cyclShift 𝑁) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 {cab 2608 ∃wrex 2913 Vcvv 3200 ∅c0 3915 ifcif 4086 〈cop 4183 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 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 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-csh 13535 |
This theorem is referenced by: cshw0 13540 cshwmodn 13541 cshwn 13543 cshwlen 13545 repswcshw 13558 |
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