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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 2820 | . . . . . . . . . . . 12 ⊢ (𝑊 = ∅ → (𝑊 ∈ V ↔ ∅ ∈ V)) | |
4 | 2, 3 | mpbiri 258 | . . . . . . . . . . 11 ⊢ (𝑊 = ∅ → 𝑊 ∈ V) |
5 | hasheq0 14330 | . . . . . . . . . . . 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 7428 | . . . . . . . 8 ⊢ (𝑊 = ∅ → (0..^(♯‘𝑊)) = (0..^0)) |
10 | fzo0 13663 | . . . . . . . 8 ⊢ (0..^0) = ∅ | |
11 | 9, 10 | eqtrdi 2787 | . . . . . . 7 ⊢ (𝑊 = ∅ → (0..^(♯‘𝑊)) = ∅) |
12 | 11 | rexeqdv 3325 | . . . . . 6 ⊢ (𝑊 = ∅ → (∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)) |
13 | 12 | rabbidv 3439 | . . . . 5 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤}) |
14 | rex0 4357 | . . . . . . . 8 ⊢ ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤 | |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ (𝑊 = ∅ → ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
16 | 15 | ralrimivw 3149 | . . . . . 6 ⊢ (𝑊 = ∅ → ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤) |
17 | rabeq0 4384 | . . . . . 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 2771 | . . . 4 ⊢ (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = ∅) |
20 | 1, 19 | eqtrid 2783 | . . 3 ⊢ (𝑊 = ∅ → 𝑀 = ∅) |
21 | 20 | fveq2d 6895 | . 2 ⊢ (𝑊 = ∅ → (♯‘𝑀) = (♯‘∅)) |
22 | hash0 14334 | . 2 ⊢ (♯‘∅) = 0 | |
23 | 21, 22 | eqtrdi 2787 | 1 ⊢ (𝑊 = ∅ → (♯‘𝑀) = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ∅c0 4322 ‘cfv 6543 (class class class)co 7412 0cc0 11116 ..^cfzo 13634 ♯chash 14297 Word cword 14471 cyclShift ccsh 14745 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 |
This theorem is referenced by: (None) |
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