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Theorem erclwwlkneqlen 26352
 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 Alexander van der Vekens, 14-Jun-2018.)
Hypotheses
Ref Expression
erclwwlkn.w 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
erclwwlkn.r = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
Assertion
Ref Expression
erclwwlkneqlen ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 → (#‘𝑇) = (#‘𝑈)))
Distinct variable groups:   𝑡,𝐸,𝑢   𝑡,𝑁,𝑢   𝑛,𝑉,𝑡,𝑢   𝑡,𝑊,𝑢   𝑇,𝑛,𝑡,𝑢   𝑈,𝑛,𝑡,𝑢   𝑛,𝑊   𝑛,𝑋   𝑛,𝑌
Allowed substitution hints:   (𝑢,𝑡,𝑛)   𝐸(𝑛)   𝑁(𝑛)   𝑋(𝑢,𝑡)   𝑌(𝑢,𝑡)

Proof of Theorem erclwwlkneqlen
StepHypRef Expression
1 erclwwlkn.w . . 3 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
2 erclwwlkn.r . . 3 = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
31, 2erclwwlkneq 26351 . 2 ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
4 fveq2 6103 . . . . . . . . 9 (𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘(𝑈 cyclShift 𝑛)))
5 clwwlknprop 26300 . . . . . . . . . . . . . 14 (𝑈 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)))
65, 1eleq2s 2706 . . . . . . . . . . . . 13 (𝑈𝑊 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)))
7 simpl2 1058 . . . . . . . . . . . . . . 15 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) ∧ 𝑛 ∈ (0...𝑁)) → 𝑈 ∈ Word 𝑉)
8 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑁 = (#‘𝑈) → (0...𝑁) = (0...(#‘𝑈)))
98eleq2d 2673 . . . . . . . . . . . . . . . . . . 19 (𝑁 = (#‘𝑈) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
109eqcoms 2618 . . . . . . . . . . . . . . . . . 18 ((#‘𝑈) = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
1110adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
12113ad2ant3 1077 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
1312biimpa 500 . . . . . . . . . . . . . . 15 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) ∧ 𝑛 ∈ (0...𝑁)) → 𝑛 ∈ (0...(#‘𝑈)))
147, 13jca 553 . . . . . . . . . . . . . 14 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈))))
1514ex 449 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈)))))
166, 15syl 17 . . . . . . . . . . . 12 (𝑈𝑊 → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈)))))
1716ad2antlr 759 . . . . . . . . . . 11 (((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈)))))
1817imp 444 . . . . . . . . . 10 ((((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈))))
19 elfzelz 12213 . . . . . . . . . . 11 (𝑛 ∈ (0...(#‘𝑈)) → 𝑛 ∈ ℤ)
20 cshwlen 13396 . . . . . . . . . . 11 ((𝑈 ∈ Word 𝑉𝑛 ∈ ℤ) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
2119, 20sylan2 490 . . . . . . . . . 10 ((𝑈 ∈ Word 𝑉𝑛 ∈ (0...(#‘𝑈))) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
2218, 21syl 17 . . . . . . . . 9 ((((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
234, 22sylan9eqr 2666 . . . . . . . 8 (((((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑇 = (𝑈 cyclShift 𝑛)) → (#‘𝑇) = (#‘𝑈))
2423ex 449 . . . . . . 7 ((((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈)))
2524rexlimdva 3013 . . . . . 6 (((𝑇𝑊𝑈𝑊) ∧ (𝑇𝑋𝑈𝑌)) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈)))
2625ex 449 . . . . 5 ((𝑇𝑊𝑈𝑊) → ((𝑇𝑋𝑈𝑌) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈))))
2726com23 84 . . . 4 ((𝑇𝑊𝑈𝑊) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → ((𝑇𝑋𝑈𝑌) → (#‘𝑇) = (#‘𝑈))))
28273impia 1253 . . 3 ((𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)) → ((𝑇𝑋𝑈𝑌) → (#‘𝑇) = (#‘𝑈)))
2928com12 32 . 2 ((𝑇𝑋𝑈𝑌) → ((𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)) → (#‘𝑇) = (#‘𝑈)))
303, 29sylbid 229 1 ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 → (#‘𝑇) = (#‘𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  {copab 4642  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalksN cclwwlkn 26277 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-rep 4699  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  ax-pre-sup 9893 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-rmo 2904  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-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  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-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  erclwwlknsym  26354  erclwwlkntr  26355
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