Theorem clwwnisshclwwn 26337

Description: Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.)

Assertion

Ref	Expression
clwwnisshclwwn	⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))

Proof of Theorem clwwnisshclwwn

Step	Hyp	Ref	Expression
1		clwwlknprop 26300	. . . . . . 7 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)))
2		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁) → 𝑁 ∈ ℕ0)
3	2	anim2i 591	. . . . . . . . . 10 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
4		df-3an 1033	. . . . . . . . . 10 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
5	3, 4	sylibr 223	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
6	5	3adant2 1073	. . . . . . . 8 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
7		clwwlkisclwwlkn 26319	. . . . . . . 8 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑊 ∈ (𝑉 ClWWalks 𝐸)))
8	6, 7	syl 17	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑊 ∈ (𝑉 ClWWalks 𝐸)))
9	1, 8	mpcom 37	. . . . . 6 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
10	9	adantl 481	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
11	10	adantr 480	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
12		eqcom 2617	. . . . . . . . . . . 12 ⊢ ((#‘𝑊) = 𝑁 ↔ 𝑁 = (#‘𝑊))
13	12	biimpi 205	. . . . . . . . . . 11 ⊢ ((#‘𝑊) = 𝑁 → 𝑁 = (#‘𝑊))
14	13	adantl 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁) → 𝑁 = (#‘𝑊))
15	14	3ad2ant3 1077	. . . . . . . . 9 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → 𝑁 = (#‘𝑊))
16	1, 15	syl 17	. . . . . . . 8 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑁 = (#‘𝑊))
17	16	adantl 481	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑁 = (#‘𝑊))
18	17	oveq2d 6565	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (0...𝑁) = (0...(#‘𝑊)))
19	18	eleq2d 2673	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
20	19	biimpa 500	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ (0...(#‘𝑊)))
21		clwwisshclwwn 26336	. . . 4 ⊢ ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑀 ∈ (0...(#‘𝑊))) → (𝑊 cyclShift 𝑀) ∈ (𝑉 ClWWalks 𝐸))
22	11, 20, 21	syl2anc 691	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑉 ClWWalks 𝐸))
23		elfzelz 12213	. . . . . . . . . 10 ⊢ (𝑀 ∈ (0...(#‘𝑊)) → 𝑀 ∈ ℤ)
24		cshwlen 13396	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
25	23, 24	sylan2 490	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑀 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
26	25	ex 449	. . . . . . . 8 ⊢ (𝑊 ∈ Word 𝑉 → (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
27	26	3ad2ant2 1076	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
28		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑁 = (#‘𝑊) → (0...𝑁) = (0...(#‘𝑊)))
29	28	eleq2d 2673	. . . . . . . . . . 11 ⊢ (𝑁 = (#‘𝑊) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
30		eqeq2 2621	. . . . . . . . . . 11 ⊢ (𝑁 = (#‘𝑊) → ((#‘(𝑊 cyclShift 𝑀)) = 𝑁 ↔ (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
31	29, 30	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑁 = (#‘𝑊) → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
32	31	eqcoms 2618	. . . . . . . . 9 ⊢ ((#‘𝑊) = 𝑁 → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁) → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
34	33	3ad2ant3 1077	. . . . . . 7 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
35	27, 34	mpbird 246	. . . . . 6 ⊢ (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
36	1, 35	syl 17	. . . . 5 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
37	36	adantl 481	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
38	37	imp 444	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁)
39	1	simp1d 1066	. . . . . . . 8 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
40	39	anim1i 590	. . . . . . 7 ⊢ ((𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ 𝑁 ∈ ℕ0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
41	40, 4	sylibr 223	. . . . . 6 ⊢ ((𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
42	41	ancoms 468	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
43	42	adantr 480	. . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
44		isclwwlkn 26297	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0) → ((𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑊 cyclShift 𝑀) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁)))
45	43, 44	syl 17	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑊 cyclShift 𝑀) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁)))
46	22, 38, 45	mpbir2and 959	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
47	46	ex 449	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  0cc0 9815  ℕ₀cn0 11169  ℤcz 11254  ...cfz 12197  #chash 12979  Word cword 13146   cyclShift ccsh 13385   ClWWalks cclwwlk 26276   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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-substr 13158  df-csh 13386  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  clwwlknscsh  26347