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Theorem clwwlkgt0 26065
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 26064 . . . 4 (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))
3 lencl 13125 . . . . . 6 (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ0)
4 nn0re 11148 . . . . . . . . 9 ((#‘𝑃) ∈ ℕ0 → (#‘𝑃) ∈ ℝ)
5 2re 10937 . . . . . . . . . 10 2 ∈ ℝ
65a1i 11 . . . . . . . . 9 ((#‘𝑃) ∈ ℕ0 → 2 ∈ ℝ)
74, 6ltnled 10035 . . . . . . . 8 ((#‘𝑃) ∈ ℕ0 → ((#‘𝑃) < 2 ↔ ¬ 2 ≤ (#‘𝑃)))
8 nn0lt2 11273 . . . . . . . . . 10 (((#‘𝑃) ∈ ℕ0 ∧ (#‘𝑃) < 2) → ((#‘𝑃) = 0 ∨ (#‘𝑃) = 1))
9 usgrav 25633 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
10 isclwwlk 26062 . . . . . . . . . . . . . . 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 13150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ Word 𝑉 → ( lastS ‘𝑃) = (𝑃‘((#‘𝑃) − 1)))
12 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = 0 → ((#‘𝑃) − 1) = (0 − 1))
1312fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑃) = 0 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(0 − 1)))
1411, 13sylan9eq 2663 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ( lastS ‘𝑃) = (𝑃‘(0 − 1)))
1514preq1d 4217 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {( lastS ‘𝑃), (𝑃‘0)} = {(𝑃‘(0 − 1)), (𝑃‘0)})
16 hasheq0 12967 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 0 ↔ 𝑃 = ∅))
17 fveq1 6087 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 = ∅ → (𝑃‘(0 − 1)) = (∅‘(0 − 1)))
18 fveq1 6087 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑃 = ∅ → (𝑃‘0) = (∅‘0))
1917, 18preq12d 4219 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {(∅‘(0 − 1)), (∅‘0)})
20 0fv 6122 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅‘(0 − 1)) = ∅
21 0fv 6122 . . . . . . . . . . . . . . . . . . . . . . . 24 (∅‘0) = ∅
2220, 21preq12i 4216 . . . . . . . . . . . . . . . . . . . . . . 23 {(∅‘(0 − 1)), (∅‘0)} = {∅, ∅}
2319, 22syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
2416, 23syl6bi 241 . . . . . . . . . . . . . . . . . . . . 21 (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 0 → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅}))
2524imp 443 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
2615, 25eqtrd 2643 . . . . . . . . . . . . . . . . . . 19 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {( lastS ‘𝑃), (𝑃‘0)} = {∅, ∅})
2726eleq1d 2671 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 ↔ {∅, ∅} ∈ ran 𝐸))
28 eqid 2609 . . . . . . . . . . . . . . . . . . . 20 ∅ = ∅
29 usgraedgrn 25676 . . . . . . . . . . . . . . . . . . . 20 ((𝑉 USGrph 𝐸 ∧ {∅, ∅} ∈ ran 𝐸) → ∅ ≠ ∅)
30 eqneqall 2792 . . . . . . . . . . . . . . . . . . . 20 (∅ = ∅ → (∅ ≠ ∅ → 2 ≤ (#‘𝑃)))
3128, 29, 30mpsyl 65 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ {∅, ∅} ∈ ran 𝐸) → 2 ≤ (#‘𝑃))
3231expcom 449 . . . . . . . . . . . . . . . . . 18 ({∅, ∅} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))
3327, 32syl6bi 241 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
3433impancom 454 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
35343adant2 1072 . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))))
3736com24 92 . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . . . . . . 19 (𝑃 ∈ Word 𝑉 → {( lastS ‘𝑃), (𝑃‘0)} = {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)})
4140eleq1d 2671 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 ↔ {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸))
42 oveq1 6534 . . . . . . . . . . . . . . . . . . . . . . . 24 ((#‘𝑃) = 1 → ((#‘𝑃) − 1) = (1 − 1))
4342fveq2d 6092 . . . . . . . . . . . . . . . . . . . . . . 23 ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(1 − 1)))
44 1m1e0 10936 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 − 1) = 0
4544fveq2i 6091 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑃‘(1 − 1)) = (𝑃‘0)
4643, 45syl6eq 2659 . . . . . . . . . . . . . . . . . . . . . 22 ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘0))
4746preq1d 4217 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑃) = 1 → {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} = {(𝑃‘0), (𝑃‘0)})
4847eleq1d 2671 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑃) = 1 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 ↔ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸))
49 eqid 2609 . . . . . . . . . . . . . . . . . . . . . 22 (𝑃‘0) = (𝑃‘0)
50 usgraedgrn 25676 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸) → (𝑃‘0) ≠ (𝑃‘0))
51 eqneqall 2792 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃‘0) = (𝑃‘0) → ((𝑃‘0) ≠ (𝑃‘0) → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃))))
5249, 50, 51mpsyl 65 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸) → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))
5352expcom 449 . . . . . . . . . . . . . . . . . . . 20 ({(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃))))
5448, 53syl6bi 241 . . . . . . . . . . . . . . . . . . 19 ((#‘𝑃) = 1 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))))
5554com14 93 . . . . . . . . . . . . . . . . . 18 (𝑃 ∈ Word 𝑉 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
5641, 55sylbid 228 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Word 𝑉 → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
5756imp 443 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
58573adant2 1072 . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . 14 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
6059com23 83 . . . . . . . . . . . . 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 84 . . . . . . . . . . 11 ((#‘𝑃) = 1 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
6339, 62jaoi 392 . . . . . . . . . 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 448 . . . . . . . 8 ((#‘𝑃) ∈ ℕ0 → ((#‘𝑃) < 2 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
667, 65sylbird 248 . . . . . . 7 ((#‘𝑃) ∈ ℕ0 → (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
6766com24 92 . . . . . 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 1076 . . . 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 85 . 2 (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
721, 71pm2.61i 174 1 (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  Vcvv 3172  c0 3873  {cpr 4126   class class class wbr 4577  ran crn 5029  cfv 5790  (class class class)co 6527  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  2c2 10917  0cn0 11139  ..^cfzo 12289  #chash 12934  Word cword 13092   lastS clsw 13093   USGrph cusg 25625   ClWWalks cclwwlk 26042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-lsw 13101  df-usgra 25628  df-clwwlk 26045
This theorem is referenced by:  clwwlkn0  26068
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