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Theorem clwwlkprop 26298
 Description: Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
Assertion
Ref Expression
clwwlkprop (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))

Proof of Theorem clwwlkprop
Dummy variables 𝑒 𝑖 𝑣 𝑤 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-clwwlk 26279 . . 3 ClWWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝑒)})
21elmpt2cl 6774 . 2 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 pm3.22 464 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉))
4 df-3an 1033 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) ↔ ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉))
53, 4sylibr 223 . . . . 5 ((𝑃 ∈ Word 𝑉 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
65a1d 25 . . . 4 ((𝑃 ∈ Word 𝑉 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V)) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
76ex 449 . . 3 (𝑃 ∈ Word 𝑉 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))))
8 clwwlk 26294 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 ClWWalks 𝐸) = {𝑝 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸)})
98eleq2d 2673 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑃 ∈ {𝑝 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸)}))
10 fveq2 6103 . . . . . . . . . . 11 (𝑝 = 𝑃 → (#‘𝑝) = (#‘𝑃))
1110oveq1d 6564 . . . . . . . . . 10 (𝑝 = 𝑃 → ((#‘𝑝) − 1) = ((#‘𝑃) − 1))
1211oveq2d 6565 . . . . . . . . 9 (𝑝 = 𝑃 → (0..^((#‘𝑝) − 1)) = (0..^((#‘𝑃) − 1)))
13 fveq1 6102 . . . . . . . . . . 11 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
14 fveq1 6102 . . . . . . . . . . 11 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
1513, 14preq12d 4220 . . . . . . . . . 10 (𝑝 = 𝑃 → {(𝑝𝑖), (𝑝‘(𝑖 + 1))} = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
1615eleq1d 2672 . . . . . . . . 9 (𝑝 = 𝑃 → ({(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))
1712, 16raleqbidv 3129 . . . . . . . 8 (𝑝 = 𝑃 → (∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))
18 fveq2 6103 . . . . . . . . . 10 (𝑝 = 𝑃 → ( lastS ‘𝑝) = ( lastS ‘𝑃))
19 fveq1 6102 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
2018, 19preq12d 4220 . . . . . . . . 9 (𝑝 = 𝑃 → {( lastS ‘𝑝), (𝑝‘0)} = {( lastS ‘𝑃), (𝑃‘0)})
2120eleq1d 2672 . . . . . . . 8 (𝑝 = 𝑃 → ({( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸 ↔ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸))
2217, 21anbi12d 743 . . . . . . 7 (𝑝 = 𝑃 → ((∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸) ↔ (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
2322elrab 3331 . . . . . 6 (𝑃 ∈ {𝑝 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸)} ↔ (𝑃 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
24 pm2.24 120 . . . . . . 7 (𝑃 ∈ Word 𝑉 → (¬ 𝑃 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
2524adantr 480 . . . . . 6 ((𝑃 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)) → (¬ 𝑃 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
2623, 25sylbi 206 . . . . 5 (𝑃 ∈ {𝑝 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑝) − 1)){(𝑝𝑖), (𝑝‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑝), (𝑝‘0)} ∈ ran 𝐸)} → (¬ 𝑃 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
279, 26syl6bi 242 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (¬ 𝑃 ∈ Word 𝑉 → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))))
2827com3r 85 . . 3 𝑃 ∈ Word 𝑉 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))))
297, 28pm2.61i 175 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉)))
302, 29mpcom 37 1 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  {cpr 4127  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   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 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-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-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  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-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-clwwlk 26279 This theorem is referenced by:  clwwlkgt0  26299  clwwlknprop  26300  clwwlkn0  26302  clwwisshclww  26335  clwwisshclwwn  26336  erclwwlkeqlen  26340  erclwwlkref  26341  erclwwlksym  26342  erclwwlktr  26343
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