Theorem clwlkclwwlk2 26904

Description: A closed walk corresponds to a closed walk as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.)

Hypotheses

Ref	Expression
clwlkclwwlk.v	⊢ 𝑉 = (Vtx‘𝐺)
clwlkclwwlk.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
clwlkclwwlk2	⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))

Distinct variable groups:   𝑓,𝐸   𝑃,𝑓   𝑓,𝑉   𝑓,𝐺

Proof of Theorem clwlkclwwlk2

Step	Hyp	Ref	Expression
1		lswccats1fst 13412	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))
2	1	3adant1 1079	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))
3	2	biantrurd 529	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalks‘𝐺) ↔ (( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalks‘𝐺))))
4		simpl 473	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 ∈ Word 𝑉)
5		wrdsymb1 13342	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃‘0) ∈ 𝑉)
6		wrdlenccats1lenm1 13399	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (𝑃‘0) ∈ 𝑉) → (#‘𝑃) = ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1))
7	4, 5, 6	syl2anc 693	. . . . . . . 8 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝑃) = ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1))
8	7	eqcomd 2628	. . . . . . 7 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1) = (#‘𝑃))
9	8	opeq2d 4409	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩ = ⟨0, (#‘𝑃)⟩)
10	9	oveq2d 6666	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩))
11	5	s1cld 13383	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉)
12		eqidd 2623	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝑃) = (#‘𝑃))
13		swrdccatid 13497	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉 ∧ (#‘𝑃) = (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩) = 𝑃)
14	4, 11, 12, 13	syl3anc 1326	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩) = 𝑃)
15	10, 14	eqtr2d 2657	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩))
16	15	3adant1 1079	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩))
17	16	eleq1d 2686	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ (ClWWalks‘𝐺) ↔ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalks‘𝐺)))
18		simp1 1061	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝐺 ∈ USPGraph )
19		simp2 1062	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 ∈ Word 𝑉)
20	11	3adant1 1079	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉)
21		ccatcl 13359	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
22	19, 20, 21	syl2anc 693	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
23		lencl 13324	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ0)
24		1e2m1 11136	. . . . . . . . . . 11 ⊢ 1 = (2 − 1)
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 1 = (2 − 1))
26	25	breq1d 4663	. . . . . . . . 9 ⊢ ((#‘𝑃) ∈ ℕ0 → (1 ≤ (#‘𝑃) ↔ (2 − 1) ≤ (#‘𝑃)))
27		2re 11090	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 2 ∈ ℝ)
29		1red 10055	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 1 ∈ ℝ)
30		nn0re 11301	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → (#‘𝑃) ∈ ℝ)
31	28, 29, 30	lesubaddd 10624	. . . . . . . . 9 ⊢ ((#‘𝑃) ∈ ℕ0 → ((2 − 1) ≤ (#‘𝑃) ↔ 2 ≤ ((#‘𝑃) + 1)))
32	26, 31	bitrd 268	. . . . . . . 8 ⊢ ((#‘𝑃) ∈ ℕ0 → (1 ≤ (#‘𝑃) ↔ 2 ≤ ((#‘𝑃) + 1)))
33	23, 32	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ Word 𝑉 → (1 ≤ (#‘𝑃) ↔ 2 ≤ ((#‘𝑃) + 1)))
34	33	biimpa 501	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + 1))
35		s1len 13385	. . . . . . 7 ⊢ (#‘⟨“(𝑃‘0)”⟩) = 1
36	35	oveq2i 6661	. . . . . 6 ⊢ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)) = ((#‘𝑃) + 1)
37	34, 36	syl6breqr 4695	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
38	37	3adant1 1079	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
39	4, 11	jca 554	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉))
40	39	3adant1 1079	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉))
41		ccatlen 13360	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉) → (#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
42	40, 41	syl 17	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
43	38, 42	breqtrrd 4681	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ (#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)))
44		clwlkclwwlk.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
45		clwlkclwwlk.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
46	44, 45	clwlkclwwlk 26903	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉 ∧ 2 ≤ (#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩))) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ (( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalks‘𝐺))))
47	18, 22, 43, 46	syl3anc 1326	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ (( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalks‘𝐺))))
48	3, 17, 47	3bitr4rd 301	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ⟨cop 4183   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   ≤ cle 10075   − cmin 10266  2c2 11070  ℕ₀cn0 11292  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294   substr csubstr 13295  Vtxcvtx 25874  iEdgciedg 25875   USPGraph cuspgr 26043  ClWalkscclwlks 26666  ClWWalkscclwwlks 26875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-wlks 26495  df-clwlks 26667  df-clwwlks 26877

This theorem is referenced by:  clwlksfoclwwlk  26963