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Theorem clwwlkgt0 26299
 Description: A closed walk in an undirected graph has a length of at least 2. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
Assertion
Ref Expression
clwwlkgt0 (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))

Proof of Theorem clwwlkgt0
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 2a1 28 . 2 (2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
2 clwwlkprop 26298 . . . 4 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
3 lencl 13179 . . . . . 6 (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ0)
4 nn0re 11178 . . . . . . . . 9 ((#‘𝑃) ∈ ℕ0 → (#‘𝑃) ∈ ℝ)
5 2re 10967 . . . . . . . . . 10 2 ∈ ℝ
65a1i 11 . . . . . . . . 9 ((#‘𝑃) ∈ ℕ0 → 2 ∈ ℝ)
74, 6ltnled 10063 . . . . . . . 8 ((#‘𝑃) ∈ ℕ0 → ((#‘𝑃) < 2 ↔ ¬ 2 ≤ (#‘𝑃)))
8 nn0lt2 11317 . . . . . . . . . 10 (((#‘𝑃) ∈ ℕ0 ∧ (#‘𝑃) < 2) → ((#‘𝑃) = 0 ∨ (#‘𝑃) = 1))
9 usgrav 25867 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
10 isclwwlk 26296 . . . . . . . . . . . . . . 15 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)))
11 lsw 13204 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ Word 𝑉 → ( lastS ‘𝑃) = (𝑃‘((#‘𝑃) − 1)))
12 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = 0 → ((#‘𝑃) − 1) = (0 − 1))
1312fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑃) = 0 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(0 − 1)))
1411, 13sylan9eq 2664 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ( lastS ‘𝑃) = (𝑃‘(0 − 1)))
1514preq1d 4218 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {( lastS ‘𝑃), (𝑃‘0)} = {(𝑃‘(0 − 1)), (𝑃‘0)})
16 hasheq0 13015 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 0 ↔ 𝑃 = ∅))
17 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 = ∅ → (𝑃‘(0 − 1)) = (∅‘(0 − 1)))
18 fveq1 6102 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 = ∅ → (𝑃‘0) = (∅‘0))
1917, 18preq12d 4220 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {(∅‘(0 − 1)), (∅‘0)})
20 0fv 6137 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅‘(0 − 1)) = ∅
21 0fv 6137 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅‘0) = ∅
2220, 21preq12i 4217 . . . . . . . . . . . . . . . . . . . . . . 23 {(∅‘(0 − 1)), (∅‘0)} = {∅, ∅}
2319, 22syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
2416, 23syl6bi 242 . . . . . . . . . . . . . . . . . . . . 21 (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 0 → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅}))
2524imp 444 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
2615, 25eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {( lastS ‘𝑃), (𝑃‘0)} = {∅, ∅})
2726eleq1d 2672 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 ↔ {∅, ∅} ∈ ran 𝐸))
28 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 ∅ = ∅
29 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {∅, ∅} ∈ ran 𝐸) → ∅ ≠ ∅)
30 eqneqall 2793 . . . . . . . . . . . . . . . . . . . 20 (∅ = ∅ → (∅ ≠ ∅ → 2 ≤ (#‘𝑃)))
3128, 29, 30mpsyl 66 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ {∅, ∅} ∈ ran 𝐸) → 2 ≤ (#‘𝑃))
3231expcom 450 . . . . . . . . . . . . . . . . . 18 ({∅, ∅} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))
3327, 32syl6bi 242 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
3433impancom 455 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
35343adant2 1073 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
3610, 35syl6bi 242 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))))
3736com24 93 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 0 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
389, 37mpcom 37 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → ((#‘𝑃) = 0 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
3938com12 32 . . . . . . . . . . 11 ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
4011preq1d 4218 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 → {( lastS ‘𝑃), (𝑃‘0)} = {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)})
4140eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 ↔ {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸))
42 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑃) = 1 → ((#‘𝑃) − 1) = (1 − 1))
4342fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(1 − 1)))
44 1m1e0 10966 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 − 1) = 0
4544fveq2i 6106 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃‘(1 − 1)) = (𝑃‘0)
4643, 45syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘0))
4746preq1d 4218 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑃) = 1 → {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} = {(𝑃‘0), (𝑃‘0)})
4847eleq1d 2672 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑃) = 1 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 ↔ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸))
49 eqid 2610 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃‘0) = (𝑃‘0)
50 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸) → (𝑃‘0) ≠ (𝑃‘0))
51 eqneqall 2793 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃‘0) = (𝑃‘0) → ((𝑃‘0) ≠ (𝑃‘0) → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃))))
5249, 50, 51mpsyl 66 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸) → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))
5352expcom 450 . . . . . . . . . . . . . . . . . . . 20 ({(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃))))
5448, 53syl6bi 242 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑃) = 1 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))))
5554com14 94 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
5641, 55sylbid 229 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
5756imp 444 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
58573adant2 1073 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
5910, 58syl6bi 242 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
6059com23 84 . . . . . . . . . . . . 13 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
619, 60mpcom 37 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
6261com3r 85 . . . . . . . . . . 11 ((#‘𝑃) = 1 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
6339, 62jaoi 393 . . . . . . . . . 10 (((#‘𝑃) = 0 ∨ (#‘𝑃) = 1) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
648, 63syl 17 . . . . . . . . 9 (((#‘𝑃) ∈ ℕ0 ∧ (#‘𝑃) < 2) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
6564ex 449 . . . . . . . 8 ((#‘𝑃) ∈ ℕ0 → ((#‘𝑃) < 2 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
667, 65sylbird 249 . . . . . . 7 ((#‘𝑃) ∈ ℕ0 → (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
6766com24 93 . . . . . 6 ((#‘𝑃) ∈ ℕ0 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
683, 67syl 17 . . . . 5 (𝑃 ∈ Word 𝑉 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
69683ad2ant3 1077 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
702, 69mpcom 37 . . 3 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃))))
7170com13 86 . 2 (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
721, 71pm2.61i 175 1 (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  2c2 10947  ℕ0cn0 11169  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   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-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-card 8648  df-cda 8873  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 This theorem is referenced by:  clwwlkn0  26302
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