Theorem clwlkclwwlk2 41212

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 13264	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))
2	1	3adant1 1072	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ( lastS ‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))
3	2	biantrurd 528	. 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 472	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 ∈ Word 𝑉)
5		wrdsymb1 13197	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃‘0) ∈ 𝑉)
6		wrdlenccats1lenm1 13252	. . . . . . . . 9 ⊢ ((𝑃 ∈ Word 𝑉 ∧ (𝑃‘0) ∈ 𝑉) → (#‘𝑃) = ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1))
7	4, 5, 6	syl2anc 691	. . . . . . . 8 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝑃) = ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1))
8	7	eqcomd 2616	. . . . . . 7 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1) = (#‘𝑃))
9	8	opeq2d 4347	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩ = ⟨0, (#‘𝑃)⟩)
10	9	oveq2d 6565	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩))
11	5	s1cld 13236	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉)
12		eqidd 2611	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝑃) = (#‘𝑃))
13		swrdccatid 13348	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉 ∧ (#‘𝑃) = (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩) = 𝑃)
14	4, 11, 12, 13	syl3anc 1318	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, (#‘𝑃)⟩) = 𝑃)
15	10, 14	eqtr2d 2645	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩))
16	15	3adant1 1072	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩))
17	16	eleq1d 2672	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ (ClWWalkS‘𝐺) ↔ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) substr ⟨0, ((#‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)⟩) ∈ (ClWWalkS‘𝐺)))
18		simp1 1054	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝐺 ∈ USPGraph )
19		simp2 1055	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃 ∈ Word 𝑉)
20	11	3adant1 1072	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉)
21		ccatcl 13212	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
22	19, 20, 21	syl2anc 691	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
23		lencl 13179	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → (#‘𝑃) ∈ ℕ0)
24		1e2m1 11013	. . . . . . . . . . 11 ⊢ 1 = (2 − 1)
25	24	a1i 11	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 1 = (2 − 1))
26	25	breq1d 4593	. . . . . . . . 9 ⊢ ((#‘𝑃) ∈ ℕ0 → (1 ≤ (#‘𝑃) ↔ (2 − 1) ≤ (#‘𝑃)))
27		2re 10967	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
28	27	a1i 11	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 2 ∈ ℝ)
29		1red 9934	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → 1 ∈ ℝ)
30		nn0re 11178	. . . . . . . . . 10 ⊢ ((#‘𝑃) ∈ ℕ0 → (#‘𝑃) ∈ ℝ)
31	28, 29, 30	lesubaddd 10503	. . . . . . . . 9 ⊢ ((#‘𝑃) ∈ ℕ0 → ((2 − 1) ≤ (#‘𝑃) ↔ 2 ≤ ((#‘𝑃) + 1)))
32	26, 31	bitrd 267	. . . . . . . 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 500	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + 1))
35		s1len 13238	. . . . . . 7 ⊢ (#‘⟨“(𝑃‘0)”⟩) = 1
36	35	oveq2i 6560	. . . . . 6 ⊢ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)) = ((#‘𝑃) + 1)
37	34, 36	syl6breqr 4625	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
38	37	3adant1 1072	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 2 ≤ ((#‘𝑃) + (#‘⟨“(𝑃‘0)”⟩)))
39	4, 11	jca 553	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉))
40	39	3adant1 1072	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉))
41		ccatlen 13213	. . . . 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 4611	. . 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 41211	. . 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 1318	. 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 300	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalkS‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalkS‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954   − cmin 10145  2c2 10947  ℕ₀cn0 11169  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  Vtxcvtx 25673  iEdgciedg 25674   USPGraph cuspgr 40378  ClWalkScclwlks 40976  ClWWalkScclwwlks 41183

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-ifp 1007  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-2o 7448  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-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-uspgr 40380  df-1wlks 40800  df-wlks 40801  df-clwlks 40977  df-clwwlks 41185

This theorem is referenced by:  clwlksfoclwwlk  41270