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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 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))

Proof of Theorem clwwnisshclwwn
StepHypRef 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)
32anim2i 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))
53, 4sylibr 223 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
653adant2 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)))
86, 7syl 17 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑊 ∈ (𝑉 ClWWalks 𝐸)))
91, 8mpcom 37 . . . . . 6 (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
109adantl 481 . . . . 5 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
1110adantr 480 . . . 4 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → 𝑊 ∈ (𝑉 ClWWalks 𝐸))
12 eqcom 2617 . . . . . . . . . . . 12 ((#‘𝑊) = 𝑁𝑁 = (#‘𝑊))
1312biimpi 205 . . . . . . . . . . 11 ((#‘𝑊) = 𝑁𝑁 = (#‘𝑊))
1413adantl 481 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁) → 𝑁 = (#‘𝑊))
15143ad2ant3 1077 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → 𝑁 = (#‘𝑊))
161, 15syl 17 . . . . . . . 8 (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑁 = (#‘𝑊))
1716adantl 481 . . . . . . 7 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → 𝑁 = (#‘𝑊))
1817oveq2d 6565 . . . . . 6 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (0...𝑁) = (0...(#‘𝑊)))
1918eleq2d 2673 . . . . 5 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
2019biimpa 500 . . . 4 (((𝑁 ∈ ℕ0𝑊 ∈ ((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))
2211, 20, 21syl2anc 691 . . 3 (((𝑁 ∈ ℕ0𝑊 ∈ ((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 𝑉𝑀 ∈ ℤ) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
2523, 24sylan2 490 . . . . . . . . 9 ((𝑊 ∈ Word 𝑉𝑀 ∈ (0...(#‘𝑊))) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))
2625ex 449 . . . . . . . 8 (𝑊 ∈ Word 𝑉 → (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
27263ad2ant2 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)))
2928eleq2d 2673 . . . . . . . . . . 11 (𝑁 = (#‘𝑊) → (𝑀 ∈ (0...𝑁) ↔ 𝑀 ∈ (0...(#‘𝑊))))
30 eqeq2 2621 . . . . . . . . . . 11 (𝑁 = (#‘𝑊) → ((#‘(𝑊 cyclShift 𝑀)) = 𝑁 ↔ (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊)))
3129, 30imbi12d 333 . . . . . . . . . 10 (𝑁 = (#‘𝑊) → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
3231eqcoms 2618 . . . . . . . . 9 ((#‘𝑊) = 𝑁 → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
3332adantl 481 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁) → ((𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁) ↔ (𝑀 ∈ (0...(#‘𝑊)) → (#‘(𝑊 cyclShift 𝑀)) = (#‘𝑊))))
34333ad2ant3 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))))
3527, 34mpbird 246 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑊 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑊) = 𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
361, 35syl 17 . . . . 5 (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
3736adantl 481 . . . 4 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁))
3837imp 444 . . 3 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (#‘(𝑊 cyclShift 𝑀)) = 𝑁)
391simp1d 1066 . . . . . . . 8 (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4039anim1i 590 . . . . . . 7 ((𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ 𝑁 ∈ ℕ0) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑁 ∈ ℕ0))
4140, 4sylibr 223 . . . . . 6 ((𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ 𝑁 ∈ ℕ0) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
4241ancoms 468 . . . . 5 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑁 ∈ ℕ0))
4342adantr 480 . . . 4 (((𝑁 ∈ ℕ0𝑊 ∈ ((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)))
4543, 44syl 17 . . 3 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ ((𝑊 cyclShift 𝑀) ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘(𝑊 cyclShift 𝑀)) = 𝑁)))
4622, 38, 45mpbir2and 959 . 2 (((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁))
4746ex 449 1 ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))
 Colors of variables: wff setvar class 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
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