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Theorem clwlkclwwlk2 29256

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.) (Proof shortened by AV, 2-Nov-2022.)

**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		simp1 1137	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝐺 ∈ USPGraph)
2		wrdsymb1 14503	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (𝑃‘0) ∈ 𝑉)
3	2	s1cld 14553	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉)
4		ccatcl 14524	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
5	3, 4	syldan 592	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
6	5	3adant1 1131	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ Word 𝑉)
7		lencl 14483	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → (♯‘𝑃) ∈ ℕ₀)
8		1e2m1 12339	. . . . . . . . . 10 ⊢ 1 = (2 − 1)
9	8	breq1i 5156	. . . . . . . . 9 ⊢ (1 ≤ (♯‘𝑃) ↔ (2 − 1) ≤ (♯‘𝑃))
10		2re 12286	. . . . . . . . . . 11 ⊢ 2 ∈ ℝ
11	10	a1i 11	. . . . . . . . . 10 ⊢ ((♯‘𝑃) ∈ ℕ₀ → 2 ∈ ℝ)
12		1red 11215	. . . . . . . . . 10 ⊢ ((♯‘𝑃) ∈ ℕ₀ → 1 ∈ ℝ)
13		nn0re 12481	. . . . . . . . . 10 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℝ)
14	11, 12, 13	lesubaddd 11811	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((2 − 1) ≤ (♯‘𝑃) ↔ 2 ≤ ((♯‘𝑃) + 1)))
15	9, 14	bitrid 283	. . . . . . . 8 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 ≤ (♯‘𝑃) ↔ 2 ≤ ((♯‘𝑃) + 1)))
16	7, 15	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ Word 𝑉 → (1 ≤ (♯‘𝑃) ↔ 2 ≤ ((♯‘𝑃) + 1)))
17	16	biimpa 478	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 2 ≤ ((♯‘𝑃) + 1))
18		s1len 14556	. . . . . . 7 ⊢ (♯‘⟨“(𝑃‘0)”⟩) = 1
19	18	oveq2i 7420	. . . . . 6 ⊢ ((♯‘𝑃) + (♯‘⟨“(𝑃‘0)”⟩)) = ((♯‘𝑃) + 1)
20	17, 19	breqtrrdi 5191	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 2 ≤ ((♯‘𝑃) + (♯‘⟨“(𝑃‘0)”⟩)))
21		ccatlen 14525	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉) → (♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((♯‘𝑃) + (♯‘⟨“(𝑃‘0)”⟩)))
22	3, 21	syldan 592	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((♯‘𝑃) + (♯‘⟨“(𝑃‘0)”⟩)))
23	20, 22	breqtrrd 5177	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 2 ≤ (♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)))
24	23	3adant1 1131	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 2 ≤ (♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)))
25		clwlkclwwlk.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
26		clwlkclwwlk.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
27	25, 26	clwlkclwwlk 29255	. . 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)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺))))
28	1, 6, 24, 27	syl3anc 1372	. 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)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺))))
29		wrdlenccats1lenm1 14572	. . . . . . . 8 ⊢ (𝑃 ∈ Word 𝑉 → ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1) = (♯‘𝑃))
30	29	oveq2d 7425	. . . . . . 7 ⊢ (𝑃 ∈ Word 𝑉 → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix (♯‘𝑃)))
31	30	adantr 482	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix (♯‘𝑃)))
32		simpl 484	. . . . . . 7 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃 ∈ Word 𝑉)
33		eqidd 2734	. . . . . . 7 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘𝑃) = (♯‘𝑃))
34		pfxccatid 14691	. . . . . . 7 ⊢ ((𝑃 ∈ Word 𝑉 ∧ ⟨“(𝑃‘0)”⟩ ∈ Word 𝑉 ∧ (♯‘𝑃) = (♯‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix (♯‘𝑃)) = 𝑃)
35	32, 3, 33, 34	syl3anc 1372	. . . . . 6 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix (♯‘𝑃)) = 𝑃)
36	31, 35	eqtr2d 2774	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃 = ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)))
37	36	eleq1d 2819	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (𝑃 ∈ (ClWWalks‘𝐺) ↔ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺)))
38		lswccats1fst 14585	. . . . 5 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (lastS‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0))
39	38	biantrurd 534	. . . 4 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺) ↔ ((lastS‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺))))
40	37, 39	bitr2d 280	. . 3 ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (((lastS‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺)) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))
41	40	3adant1 1131	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (((lastS‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) = ((𝑃 ++ ⟨“(𝑃‘0)”⟩)‘0) ∧ ((𝑃 ++ ⟨“(𝑃‘0)”⟩) prefix ((♯‘(𝑃 ++ ⟨“(𝑃‘0)”⟩)) − 1)) ∈ (ClWWalks‘𝐺)) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))
42	28, 41	bitrd 279	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 ≤ cle 11249 − cmin 11444 2c2 12267 ℕ₀cn0 12472 ♯chash 14290 Word cword 14464 lastSclsw 14512 ++ cconcat 14520 ⟨“cs1 14545 prefix cpfx 14620 Vtxcvtx 28256 iEdgciedg 28257 USPGraphcuspgr 28408 ClWalkscclwlks 29027 ClWWalkscclwwlk 29234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-lsw 14513 df-concat 14521 df-s1 14546 df-substr 14591 df-pfx 14621 df-edg 28308 df-uhgr 28318 df-upgr 28342 df-uspgr 28410 df-wlks 28856 df-clwlks 29028 df-clwwlk 29235

This theorem is referenced by: clwlkclwwlkfo 29262

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