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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.

Step	Hyp	Ref	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
6	5	a1i 11	. . . . . . . . 9 ⊢ ((#‘𝑃) ∈ ℕ₀ → 2 ∈ ℝ)
7	4, 6	ltnled 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)){(𝑃‘𝑖), (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))
13	12	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) = 0 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(0 − 1)))
14	11, 13	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ( lastS ‘𝑃) = (𝑃‘(0 − 1)))
15	14	preq1d 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))
19	17, 18	preq12d 4220	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {(∅‘(0 − 1)), (∅‘0)})
20		0fv 6137	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅‘(0 − 1)) = ∅
21		0fv 6137	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∅‘0) = ∅
22	20, 21	preq12i 4217	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {(∅‘(0 − 1)), (∅‘0)} = {∅, ∅}
23	19, 22	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑃 = ∅ → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
24	16, 23	syl6bi 242	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑃 ∈ Word 𝑉 → ((#‘𝑃) = 0 → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅}))
25	24	imp 444	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {(𝑃‘(0 − 1)), (𝑃‘0)} = {∅, ∅})
26	15, 25	eqtrd 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → {( lastS ‘𝑃), (𝑃‘0)} = {∅, ∅})
27	26	eleq1d 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)))
31	28, 29, 30	mpsyl 66	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑉 USGrph 𝐸 ∧ {∅, ∅} ∈ ran 𝐸) → 2 ≤ (#‘𝑃))
32	31	expcom 450	. . . . . . . . . . . . . . . . . 18 ⊢ ({∅, ∅} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))
33	27, 32	syl6bi 242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 0) → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
34	33	impancom 455	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
35	34	3adant2 1073	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃))))
36	10, 35	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → 2 ≤ (#‘𝑃)))))
37	36	com24 93	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 0 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
38	9, 37	mpcom 37	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → ((#‘𝑃) = 0 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
39	38	com12 32	. . . . . . . . . . 11 ⊢ ((#‘𝑃) = 0 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
40	11	preq1d 4218	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ Word 𝑉 → {( lastS ‘𝑃), (𝑃‘0)} = {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)})
41	40	eleq1d 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))
43	42	fveq2d 6107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘(1 − 1)))
44		1m1e0 10966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (1 − 1) = 0
45	44	fveq2i 6106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑃‘(1 − 1)) = (𝑃‘0)
46	43, 45	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) = 1 → (𝑃‘((#‘𝑃) − 1)) = (𝑃‘0))
47	46	preq1d 4218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑃) = 1 → {(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} = {(𝑃‘0), (𝑃‘0)})
48	47	eleq1d 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))))
52	49, 50, 51	mpsyl 66	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸) → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))
53	52	expcom 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({(𝑃‘0), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃))))
54	48, 53	syl6bi 242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝑃) = 1 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑃 ∈ Word 𝑉 → 2 ≤ (#‘𝑃)))))
55	54	com14 94	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Word 𝑉 → ({(𝑃‘((#‘𝑃) − 1)), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
56	41, 55	sylbid 229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ Word 𝑉 → ({( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
57	56	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ Word 𝑉 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
58	57	3adant2 1073	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
59	10, 58	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
60	59	com23 84	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃)))))
61	9, 60	mpcom 37	. . . . . . . . . . . 12 ⊢ (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → ((#‘𝑃) = 1 → 2 ≤ (#‘𝑃))))
62	61	com3r 85	. . . . . . . . . . 11 ⊢ ((#‘𝑃) = 1 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
63	39, 62	jaoi 393	. . . . . . . . . 10 ⊢ (((#‘𝑃) = 0 ∨ (#‘𝑃) = 1) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (((#‘𝑃) ∈ ℕ₀ ∧ (#‘𝑃) < 2) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
65	64	ex 449	. . . . . . . 8 ⊢ ((#‘𝑃) ∈ ℕ₀ → ((#‘𝑃) < 2 → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
66	7, 65	sylbird 249	. . . . . . 7 ⊢ ((#‘𝑃) ∈ ℕ₀ → (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))))
67	66	com24 93	. . . . . 6 ⊢ ((#‘𝑃) ∈ ℕ₀ → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
68	3, 67	syl 17	. . . . 5 ⊢ (𝑃 ∈ Word 𝑉 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
69	68	3ad2ant3 1077	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉) → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃)))))
70	2, 69	mpcom 37	. . 3 ⊢ (𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 USGrph 𝐸 → (¬ 2 ≤ (#‘𝑃) → 2 ≤ (#‘𝑃))))
71	70	com13 86	. 2 ⊢ (¬ 2 ≤ (#‘𝑃) → (𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃))))
72	1, 71	pm2.61i 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

W3C validator