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Theorem clwwlkn2 26303
 Description: In an undirected simple graph, a closed walk of length 2 represented as word is a word consisting of 2 symbols representing vertices connected by an edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
Assertion
Ref Expression
clwwlkn2 (𝑉 USGrph 𝐸 → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))

Proof of Theorem clwwlkn2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 usgrav 25867 . . 3 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2 2nn0 11186 . . . . 5 2 ∈ ℕ0
3 isclwwlkn 26297 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 2 ∈ ℕ0) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ (𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑊) = 2)))
42, 3mp3an3 1405 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ (𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑊) = 2)))
5 isclwwlk 26296 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
65anbi1d 737 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑊) = 2) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2)))
74, 6bitrd 267 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2)))
81, 7syl 17 . 2 (𝑉 USGrph 𝐸 → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2)))
9 3anass 1035 . . . . 5 ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
10 oveq1 6556 . . . . . . . . . . . 12 ((#‘𝑊) = 2 → ((#‘𝑊) − 1) = (2 − 1))
1110oveq2d 6565 . . . . . . . . . . 11 ((#‘𝑊) = 2 → (0..^((#‘𝑊) − 1)) = (0..^(2 − 1)))
1211raleqdv 3121 . . . . . . . . . 10 ((#‘𝑊) = 2 → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(2 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
1312ad2antlr 759 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ (0..^(2 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
14 2m1e1 11012 . . . . . . . . . . . . 13 (2 − 1) = 1
1514oveq2i 6560 . . . . . . . . . . . 12 (0..^(2 − 1)) = (0..^1)
16 fzo01 12417 . . . . . . . . . . . 12 (0..^1) = {0}
1715, 16eqtri 2632 . . . . . . . . . . 11 (0..^(2 − 1)) = {0}
1817a1i 11 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (0..^(2 − 1)) = {0})
1918raleqdv 3121 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^(2 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ ∀𝑖 ∈ {0} {(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))
20 c0ex 9913 . . . . . . . . . 10 0 ∈ V
21 fveq2 6103 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑊𝑖) = (𝑊‘0))
22 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
23 0p1e1 11009 . . . . . . . . . . . . . . 15 (0 + 1) = 1
2422, 23syl6eq 2660 . . . . . . . . . . . . . 14 (𝑖 = 0 → (𝑖 + 1) = 1)
2524fveq2d 6107 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑊‘(𝑖 + 1)) = (𝑊‘1))
2621, 25preq12d 4220 . . . . . . . . . . . 12 (𝑖 = 0 → {(𝑊𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘0), (𝑊‘1)})
2726eleq1d 2672 . . . . . . . . . . 11 (𝑖 = 0 → ({(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
2827ralsng 4165 . . . . . . . . . 10 (0 ∈ V → (∀𝑖 ∈ {0} {(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
2920, 28mp1i 13 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (∀𝑖 ∈ {0} {(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
3013, 19, 293bitrd 293 . . . . . . . 8 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
3130anbi1d 737 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ((∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ↔ ({(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
32 lsw 13204 . . . . . . . . . . . . . 14 (𝑊 ∈ Word 𝑉 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
3332adantl 481 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
3410ad2antlr 759 . . . . . . . . . . . . . . 15 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑊) − 1) = (2 − 1))
3534, 14syl6eq 2660 . . . . . . . . . . . . . 14 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ((#‘𝑊) − 1) = 1)
3635fveq2d 6107 . . . . . . . . . . . . 13 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘1))
3733, 36eqtr2d 2645 . . . . . . . . . . . 12 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → (𝑊‘1) = ( lastS ‘𝑊))
3837preq2d 4219 . . . . . . . . . . 11 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → {(𝑊‘0), (𝑊‘1)} = {(𝑊‘0), ( lastS ‘𝑊)})
39 prcom 4211 . . . . . . . . . . 11 {(𝑊‘0), ( lastS ‘𝑊)} = {( lastS ‘𝑊), (𝑊‘0)}
4038, 39syl6eq 2660 . . . . . . . . . 10 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → {(𝑊‘0), (𝑊‘1)} = {( lastS ‘𝑊), (𝑊‘0)})
4140eleq1d 2672 . . . . . . . . 9 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ({(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸 ↔ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸))
4241biimpd 218 . . . . . . . 8 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ({(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸 → {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸))
4342pm4.71d 664 . . . . . . 7 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ({(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸 ↔ ({(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))
4431, 43bitr4d 270 . . . . . 6 (((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) ∧ 𝑊 ∈ Word 𝑉) → ((∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
4544pm5.32da 671 . . . . 5 ((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) → ((𝑊 ∈ Word 𝑉 ∧ (∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)) ↔ (𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))
469, 45syl5bb 271 . . . 4 ((𝑉 USGrph 𝐸 ∧ (#‘𝑊) = 2) → ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))
4746ex 449 . . 3 (𝑉 USGrph 𝐸 → ((#‘𝑊) = 2 → ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))))
4847pm5.32rd 670 . 2 (𝑉 USGrph 𝐸 → (((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2) ↔ ((𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2)))
49 3anass 1035 . . . 4 (((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) ↔ ((#‘𝑊) = 2 ∧ (𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))
50 ancom 465 . . . 4 (((#‘𝑊) = 2 ∧ (𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)) ↔ ((𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2))
5149, 50bitr2i 264 . . 3 (((𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
5251a1i 11 . 2 (𝑉 USGrph 𝐸 → (((𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) ∧ (#‘𝑊) = 2) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))
538, 48, 523bitrd 293 1 (𝑉 USGrph 𝐸 → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {csn 4125  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  2c2 10947  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   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 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-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-usgra 25862  df-clwwlk 26279  df-clwwlkn 26280 This theorem is referenced by:  numclwwlkovf2  26611
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