Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cshwsdisj | Structured version Visualization version GIF version |
Description: The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.) |
Ref | Expression |
---|---|
cshwshash.0 | ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ)) |
Ref | Expression |
---|---|
cshwsdisj | ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 861 | . . . . 5 ⊢ (𝑛 = 𝑗 → (𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅)) | |
2 | 1 | a1d 25 | . . . 4 ⊢ (𝑛 = 𝑗 → (((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊)))) → (𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅))) |
3 | simprl 767 | . . . . . . . 8 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → (𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0))) | |
4 | simprrl 777 | . . . . . . . 8 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → 𝑛 ∈ (0..^(♯‘𝑊))) | |
5 | simprrr 778 | . . . . . . . 8 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → 𝑗 ∈ (0..^(♯‘𝑊))) | |
6 | necom 3066 | . . . . . . . . . 10 ⊢ (𝑛 ≠ 𝑗 ↔ 𝑗 ≠ 𝑛) | |
7 | 6 | biimpi 217 | . . . . . . . . 9 ⊢ (𝑛 ≠ 𝑗 → 𝑗 ≠ 𝑛) |
8 | 7 | adantr 481 | . . . . . . . 8 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → 𝑗 ≠ 𝑛) |
9 | cshwshash.0 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ)) | |
10 | 9 | cshwshashlem3 16419 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ((𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ≠ 𝑛) → (𝑊 cyclShift 𝑛) ≠ (𝑊 cyclShift 𝑗))) |
11 | 10 | imp 407 | . . . . . . . 8 ⊢ (((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ≠ 𝑛)) → (𝑊 cyclShift 𝑛) ≠ (𝑊 cyclShift 𝑗)) |
12 | 3, 4, 5, 8, 11 | syl13anc 1364 | . . . . . . 7 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → (𝑊 cyclShift 𝑛) ≠ (𝑊 cyclShift 𝑗)) |
13 | disjsn2 4640 | . . . . . . 7 ⊢ ((𝑊 cyclShift 𝑛) ≠ (𝑊 cyclShift 𝑗) → ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅) |
15 | 14 | olcd 870 | . . . . 5 ⊢ ((𝑛 ≠ 𝑗 ∧ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊))))) → (𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅)) |
16 | 15 | ex 413 | . . . 4 ⊢ (𝑛 ≠ 𝑗 → (((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊)))) → (𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅))) |
17 | 2, 16 | pm2.61ine 3097 | . . 3 ⊢ (((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) ∧ (𝑛 ∈ (0..^(♯‘𝑊)) ∧ 𝑗 ∈ (0..^(♯‘𝑊)))) → (𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅)) |
18 | 17 | ralrimivva 3188 | . 2 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → ∀𝑛 ∈ (0..^(♯‘𝑊))∀𝑗 ∈ (0..^(♯‘𝑊))(𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅)) |
19 | oveq2 7153 | . . . 4 ⊢ (𝑛 = 𝑗 → (𝑊 cyclShift 𝑛) = (𝑊 cyclShift 𝑗)) | |
20 | 19 | sneqd 4569 | . . 3 ⊢ (𝑛 = 𝑗 → {(𝑊 cyclShift 𝑛)} = {(𝑊 cyclShift 𝑗)}) |
21 | 20 | disjor 5037 | . 2 ⊢ (Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)} ↔ ∀𝑛 ∈ (0..^(♯‘𝑊))∀𝑗 ∈ (0..^(♯‘𝑊))(𝑛 = 𝑗 ∨ ({(𝑊 cyclShift 𝑛)} ∩ {(𝑊 cyclShift 𝑗)}) = ∅)) |
22 | 18, 21 | sylibr 235 | 1 ⊢ ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 841 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ∅c0 4288 {csn 4557 Disj wdisj 5022 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ..^cfzo 13021 ♯chash 13678 Word cword 13849 cyclShift ccsh 14138 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-concat 13911 df-substr 13991 df-pfx 14021 df-reps 14119 df-csh 14139 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-prm 16004 df-phi 16091 |
This theorem is referenced by: cshwshashnsame 16425 |
Copyright terms: Public domain | W3C validator |