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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}

Assertion

Ref	Expression
erclwwlkneqlen	⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (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

Step	Hyp	Ref	Expression
1		erclwwlkn.w	. . 3 ⊢ 𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)
2		erclwwlkn.r	. . 3 ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}
3	1, 2	erclwwlkneq 26351	. 2 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃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)))
6	5, 1	eleq2s 2706	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝑊 → ((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)))
9	8	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 = (#‘𝑈) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
10	9	eqcoms 2618	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑈) = 𝑁 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
11	10	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
12	11	3ad2ant3 1077	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ (0...(#‘𝑈))))
13	12	biimpa 500	. . . . . . . . . . . . . . 15 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) ∧ 𝑛 ∈ (0...𝑁)) → 𝑛 ∈ (0...(#‘𝑈)))
14	7, 13	jca 553	. . . . . . . . . . . . . 14 ⊢ ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈))))
15	14	ex 449	. . . . . . . . . . . . 13 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑈 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑈) = 𝑁)) → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈)))))
16	6, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝑊 → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈)))))
17	16	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) → (𝑛 ∈ (0...𝑁) → (𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈)))))
18	17	imp 444	. . . . . . . . . 10 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈))))
19		elfzelz 12213	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0...(#‘𝑈)) → 𝑛 ∈ ℤ)
20		cshwlen 13396	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ ℤ) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
21	19, 20	sylan2 490	. . . . . . . . . 10 ⊢ ((𝑈 ∈ Word 𝑉 ∧ 𝑛 ∈ (0...(#‘𝑈))) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
22	18, 21	syl 17	. . . . . . . . 9 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (#‘(𝑈 cyclShift 𝑛)) = (#‘𝑈))
23	4, 22	sylan9eqr 2666	. . . . . . . 8 ⊢ (((((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) ∧ 𝑛 ∈ (0...𝑁)) ∧ 𝑇 = (𝑈 cyclShift 𝑛)) → (#‘𝑇) = (#‘𝑈))
24	23	ex 449	. . . . . . 7 ⊢ ((((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) ∧ 𝑛 ∈ (0...𝑁)) → (𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈)))
25	24	rexlimdva 3013	. . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) ∧ (𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌)) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈)))
26	25	ex 449	. . . . 5 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) → ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → (#‘𝑇) = (#‘𝑈))))
27	26	com23 84	. . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊) → (∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛) → ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (#‘𝑇) = (#‘𝑈))))
28	27	3impia 1253	. . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)) → ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (#‘𝑇) = (#‘𝑈)))
29	28	com12 32	. 2 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → ((𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛)) → (#‘𝑇) = (#‘𝑈)))
30	3, 29	sylbid 229	1 ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (#‘𝑇) = (#‘𝑈)))

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