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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 5307 | . . . . . . . . . . . 12 ⊢ ∅ ∈ V | |
3 | eleq1 2817 | . . . . . . . . . . . 12 ⊢ (𝑊 = ∅ → (𝑊 ∈ V ↔ ∅ ∈ V)) | |
4 | 2, 3 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝑊 = ∅ → 𝑊 ∈ V) |
5 | hasheq0 14355 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ V → ((♯‘𝑊) = 0 ↔ 𝑊 = ∅)) | |
6 | 5 | bicomd 222 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ V → (𝑊 = ∅ ↔ (♯‘𝑊) = 0)) |
7 | 4, 6 | syl 17 | . . . . . . . . . 10 ⊢ (𝑊 = ∅ → (𝑊 = ∅ ↔ (♯‘𝑊) = 0)) |
8 | 7 | ibi 267 | . . . . . . . . 9 ⊢ (𝑊 = ∅ → (♯‘𝑊) = 0) |
9 | 8 | oveq2d 7436 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (0..^(♯‘𝑊)) = (0..^0)) |
10 | fzo0 13689 | . . . . . . . 8 ⊢ (0..^0) = ∅ | |
11 | 9, 10 | eqtrdi 2784 | . . . . . . 7 ⊢ (𝑊 = ∅ → (0..^(♯‘𝑊)) = ∅) |
12 | 11 | rexeqdv 3323 | . . . . . 6 ⊢ (𝑊 = ∅ → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)) |
13 | 12 | rabbidv 3437 | . . . . 5 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤}) |
14 | rex0 4358 | . . . . . . . 8 ⊢ ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑊 = ∅ → ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
16 | 15 | ralrimivw 3147 | . . . . . 6 ⊢ (𝑊 = ∅ → ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
17 | rabeq0 4385 | . . . . . 6 ⊢ ({𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) | |
18 | 16, 17 | sylibr 233 | . . . . 5 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅) |
19 | 13, 18 | eqtrd 2768 | . . . 4 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = ∅) |
20 | 1, 19 | eqtrid 2780 | . . 3 ⊢ (𝑊 = ∅ → 𝑀 = ∅) |
21 | 20 | fveq2d 6901 | . 2 ⊢ (𝑊 = ∅ → (♯‘𝑀) = (♯‘∅)) |
22 | hash0 14359 | . 2 ⊢ (♯‘∅) = 0 | |
23 | 21, 22 | eqtrdi 2784 | 1 ⊢ (𝑊 = ∅ → (♯‘𝑀) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∃wrex 3067 {crab 3429 Vcvv 3471 ∅c0 4323 ‘cfv 6548 (class class class)co 7420 0cc0 11139 ..^cfzo 13660 ♯chash 14322 Word cword 14497 cyclShift ccsh 14771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 |
This theorem is referenced by: (None) |
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