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Theorem cshws0 15646
 Description: The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
Hypothesis
Ref Expression
cshwrepswhash1.m 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
Assertion
Ref Expression
cshws0 (𝑊 = ∅ → (#‘𝑀) = 0)
Distinct variable groups:   𝑛,𝑉,𝑤   𝑛,𝑊,𝑤
Allowed substitution hints:   𝑀(𝑤,𝑛)

Proof of Theorem cshws0
StepHypRef Expression
1 cshwrepswhash1.m . . . 4 𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}
2 0ex 4718 . . . . . . . . . . . 12 ∅ ∈ V
3 eleq1 2676 . . . . . . . . . . . 12 (𝑊 = ∅ → (𝑊 ∈ V ↔ ∅ ∈ V))
42, 3mpbiri 247 . . . . . . . . . . 11 (𝑊 = ∅ → 𝑊 ∈ V)
5 hasheq0 13015 . . . . . . . . . . . 12 (𝑊 ∈ V → ((#‘𝑊) = 0 ↔ 𝑊 = ∅))
65bicomd 212 . . . . . . . . . . 11 (𝑊 ∈ V → (𝑊 = ∅ ↔ (#‘𝑊) = 0))
74, 6syl 17 . . . . . . . . . 10 (𝑊 = ∅ → (𝑊 = ∅ ↔ (#‘𝑊) = 0))
87ibi 255 . . . . . . . . 9 (𝑊 = ∅ → (#‘𝑊) = 0)
98oveq2d 6565 . . . . . . . 8 (𝑊 = ∅ → (0..^(#‘𝑊)) = (0..^0))
10 fzo0 12361 . . . . . . . 8 (0..^0) = ∅
119, 10syl6eq 2660 . . . . . . 7 (𝑊 = ∅ → (0..^(#‘𝑊)) = ∅)
1211rexeqdv 3122 . . . . . 6 (𝑊 = ∅ → (∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤 ↔ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤))
1312rabbidv 3164 . . . . 5 (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤})
14 rex0 3894 . . . . . . . 8 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤
1514a1i 11 . . . . . . 7 (𝑊 = ∅ → ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)
1615ralrimivw 2950 . . . . . 6 (𝑊 = ∅ → ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)
17 rabeq0 3911 . . . . . 6 ({𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤)
1816, 17sylibr 223 . . . . 5 (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ ∅ (𝑊 cyclShift 𝑛) = 𝑤} = ∅)
1913, 18eqtrd 2644 . . . 4 (𝑊 = ∅ → {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(#‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤} = ∅)
201, 19syl5eq 2656 . . 3 (𝑊 = ∅ → 𝑀 = ∅)
2120fveq2d 6107 . 2 (𝑊 = ∅ → (#‘𝑀) = (#‘∅))
22 hash0 13019 . 2 (#‘∅) = 0
2321, 22syl6eq 2660 1 (𝑊 = ∅ → (#‘𝑀) = 0)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980 This theorem is referenced by: (None)
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