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Mirrors > Home > MPE Home > Th. List > cshws0 | Structured version Visualization version GIF version |
Description: The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.) |
Ref | Expression |
---|---|
cshwrepswhash1.m | ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} |
Ref | Expression |
---|---|
cshws0 | ⊢ (𝑊 = ∅ → (#‘𝑀) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cshwrepswhash1.m | . . . 4 ⊢ 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} | |
2 | 0ex 4790 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
3 | eleq1 2689 | . . . . . . . . . . . 12 ⊢ (𝑊 = ∅ → (𝑊 ∈ V ↔ ∅ ∈ V)) | |
4 | 2, 3 | mpbiri 248 | . . . . . . . . . . 11 ⊢ (𝑊 = ∅ → 𝑊 ∈ V) |
5 | hasheq0 13154 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ V → ((#‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
6 | 5 | bicomd 213 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ V → (𝑊 = ∅ ↔ (#‘𝑊) = 0)) |
7 | 4, 6 | syl 17 | . . . . . . . . . 10 ⊢ (𝑊 = ∅ → (𝑊 = ∅ ↔ (#‘𝑊) = 0)) |
8 | 7 | ibi 256 | . . . . . . . . 9 ⊢ (𝑊 = ∅ → (#‘𝑊) = 0) |
9 | 8 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (0..^(#‘𝑊)) = (0..^0)) |
10 | fzo0 12492 | . . . . . . . 8 ⊢ (0..^0) = ∅ | |
11 | 9, 10 | syl6eq 2672 | . . . . . . 7 ⊢ (𝑊 = ∅ → (0..^(#‘𝑊)) = ∅) |
12 | 11 | rexeqdv 3145 | . . . . . 6 ⊢ (𝑊 = ∅ → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)) |
13 | 12 | rabbidv 3189 | . . . . 5 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤}) |
14 | rex0 3938 | . . . . . . . 8 ⊢ ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑊 = ∅ → ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
16 | 15 | ralrimivw 2967 | . . . . . 6 ⊢ (𝑊 = ∅ → ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
17 | rabeq0 3957 | . . . . . 6 ⊢ ({𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) | |
18 | 16, 17 | sylibr 224 | . . . . 5 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅) |
19 | 13, 18 | eqtrd 2656 | . . . 4 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = ∅) |
20 | 1, 19 | syl5eq 2668 | . . 3 ⊢ (𝑊 = ∅ → 𝑀 = ∅) |
21 | 20 | fveq2d 6195 | . 2 ⊢ (𝑊 = ∅ → (#‘𝑀) = (#‘∅)) |
22 | hash0 13158 | . 2 ⊢ (#‘∅) = 0 | |
23 | 21, 22 | syl6eq 2672 | 1 ⊢ (𝑊 = ∅ → (#‘𝑀) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∅c0 3915 ‘cfv 5888 (class class class)co 6650 0cc0 9936 ..^cfzo 12465 #chash 13117 Word cword 13291 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-int 4476 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-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 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-hash 13118 |
This theorem is referenced by: (None) |
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