Theorem erclwwlkeqlen 26340

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, 11-Jun-2018.)

Hypothesis

Ref	Expression
erclwwlk.r	⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkeqlen	⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊)))

Distinct variable groups:   𝑛,𝐸,𝑢,𝑤   𝑛,𝑉,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 erclwwlkeqlen

Step	Hyp	Ref	Expression
1		erclwwlk.r	. . 3 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2	1	erclwwlkeq 26339	. 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
3		fveq2 6103	. . . . . . . . 9 ⊢ (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘(𝑊 cyclShift 𝑛)))
4		clwwlkprop 26298	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉))
5		pm3.2 462	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑉 → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊)))))
6	5	3ad2ant3 1077	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ Word 𝑉) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊)))))
7	4, 6	syl 17	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ (𝑉 ClWWalks 𝐸) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊)))))
8	7	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (𝑛 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊)))))
9	8	imp 444	. . . . . . . . . 10 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))))
10		elfzelz 12213	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(#‘𝑊)) → 𝑛 ∈ ℤ)
11		cshwlen 13396	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
12	10, 11	sylan2 490	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
13	9, 12	syl 17	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
14	3, 13	sylan9eqr 2666	. . . . . . . 8 ⊢ (((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) ∧ 𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊))
15	14	ex 449	. . . . . . 7 ⊢ ((((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
16	15	rexlimdva 3013	. . . . . 6 ⊢ (((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
17	16	ex 449	. . . . 5 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊))))
18	17	com23 84	. . . 4 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊))))
19	18	3impia 1253	. . 3 ⊢ ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊)))
20	19	com12 32	. 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊)))
21	2, 20	sylbid 229	1 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊)))

Syntax hints:   → wi 4   ∧ 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  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalks cclwwlk 26276

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

This theorem is referenced by:  erclwwlksym  26342  erclwwlktr  26343