Theorem erclwwlkseqlen 26933

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	⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (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

Step	Hyp	Ref	Expression
1		erclwwlks.r	. . 3 ⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}
2	1	erclwwlkseq 26932	. 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
3		fveq2 6191	. . . . . . . . 9 ⊢ (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘(𝑊 cyclShift 𝑛)))
4		eqid 2622	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
5	4	clwwlkbp 26883	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅))
6	5	simp2d 1074	. . . . . . . . . . 11 ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 𝑊 ∈ Word (Vtx‘𝐺))
7	6	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → 𝑊 ∈ Word (Vtx‘𝐺))
8		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑛 ∈ (0...(#‘𝑊)) → 𝑛 ∈ ℤ)
9		cshwlen 13545	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 𝑛 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
10	7, 8, 9	syl2an 494	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑛)) = (#‘𝑊))
11	3, 10	sylan9eqr 2678	. . . . . . . 8 ⊢ (((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) ∧ 𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊))
12	11	ex 450	. . . . . . 7 ⊢ ((((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) ∧ 𝑛 ∈ (0...(#‘𝑊))) → (𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
13	12	rexlimdva 3031	. . . . . 6 ⊢ (((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) ∧ (𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊)))
14	13	ex 450	. . . . 5 ⊢ ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → (#‘𝑈) = (#‘𝑊))))
15	14	com23 86	. . . 4 ⊢ ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺)) → (∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊))))
16	15	3impia 1261	. . 3 ⊢ ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (#‘𝑈) = (#‘𝑊)))
17	16	com12 32	. 2 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛)) → (#‘𝑈) = (#‘𝑊)))
18	2, 17	sylbid 230	1 ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200  ∅c0 3915   class class class wbr 4653  {copab 4712  ‘cfv 5888  (class class class)co 6650  0cc0 9936  ℤcz 11377  ...cfz 12326  #chash 13117  Word cword 13291   cyclShift ccsh 13534  Vtxcvtx 25874  ClWWalkscclwwlks 26875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877

This theorem is referenced by:  erclwwlkssym  26935  erclwwlkstr  26936