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Theorem erclwwlkseqlen 26793
 Description: If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)
Hypothesis
Ref Expression
erclwwlks.r = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkseqlen ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 → (#‘𝑈) = (#‘𝑊)))
Distinct variable groups:   𝑛,𝐺,𝑢,𝑤   𝑈,𝑛,𝑢,𝑤   𝑛,𝑊,𝑢,𝑤   𝑛,𝑋   𝑛,𝑌
Allowed substitution hints:   (𝑤,𝑢,𝑛)   𝑋(𝑤,𝑢)   𝑌(𝑤,𝑢)

Proof of Theorem erclwwlkseqlen
StepHypRef Expression
1 erclwwlks.r . . 3 = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
21erclwwlkseq 26792 . 2 ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
3 fveq2 6150 . . . . . . . . 9 (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘(𝑊 cyclShift 𝑛)))
4 eqid 2626 . . . . . . . . . . . . 13 (Vtx‘𝐺) = (Vtx‘𝐺)
54clwwlkbp 26744 . . . . . . . . . . . 12 (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅))
65simp2d 1072 . . . . . . . . . . 11 (𝑊 ∈ (ClWWalks‘𝐺) → 𝑊 ∈ Word (Vtx‘𝐺))
76ad2antlr 762 . . . . . . . . . 10 (((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈𝑋𝑊𝑌)) → 𝑊 ∈ Word (Vtx‘𝐺))
8 elfzelz 12281 . . . . . . . . . 10 (𝑛 ∈ (0...(#‘𝑊)) → 𝑛 ∈ ℤ)
9 cshwlen 13477 . . . . . . . . . 10 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
107, 8, 9syl2an 494 . . . . . . . . 9 ((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈𝑋𝑊𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
113, 10sylan9eqr 2682 . . . . . . . 8 (((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈𝑋𝑊𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) ∧ 𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊))
1211ex 450 . . . . . . 7 ((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈𝑋𝑊𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
1312rexlimdva 3029 . . . . . 6 (((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈𝑋𝑊𝑌)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
1413ex 450 . . . . 5 ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) → ((𝑈𝑋𝑊𝑌) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊))))
1514com23 86 . . . 4 ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → ((𝑈𝑋𝑊𝑌) → (#‘𝑈) = (#‘𝑊))))
16153impia 1258 . . 3 ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → ((𝑈𝑋𝑊𝑌) → (#‘𝑈) = (#‘𝑊)))
1716com12 32 . 2 ((𝑈𝑋𝑊𝑌) → ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊)))
182, 17sylbid 230 1 ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 → (#‘𝑈) = (#‘𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992   ≠ wne 2796  ∃wrex 2913  Vcvv 3191  ∅c0 3896   class class class wbr 4618  {copab 4677  ‘cfv 5850  (class class class)co 6605  0cc0 9881  ℤcz 11322  ...cfz 12265  #chash 13054  Word cword 13225   cyclShift ccsh 13466  Vtxcvtx 25769  ClWWalkscclwwlks 26736 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-rp 11777  df-fz 12266  df-fzo 12404  df-fl 12530  df-mod 12606  df-hash 13055  df-word 13233  df-concat 13235  df-substr 13237  df-csh 13467  df-clwwlks 26738 This theorem is referenced by:  erclwwlkssym  26795  erclwwlkstr  26796
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